I wanted to get the community's input on an interesting combination and whether it could/should work in practice. 

The build is a Rogue taking the Phantom subclass and combining this with the Keen Mind feat. At third level, when selecting the Phantom subclass, you gain the Phantom feature Whispers of the Dead. Whispers of the Dead states, "... Whenever you finish a short or long rest, you can choose one skill or tool proficiency that you lack and gain it, as a ghostly presence shares its knowledge with you. You lose this proficiency when you use this feature to choose a different proficiency that you lack." Meanwhile the Keen Mind feat states, among other things, "You can accurately recall anything you have seen or heard within the past month." 

The idea is that a Phantom Rogue with Keen Mind could use Whispers of the Dead to gain proficiency with Acrobatics on Day 1. On Day 2, they use the feature once more to gain proficiency with Arcana, thus losing the Acrobatics proficiency. However, the Rogue would then use Keen Mind to recall the lessons from Day 1, thus regaining the Acrobatics proficiency. This would then be repeated each day on a ~30 day cycle (as per Keen Mind's one month limit on recalling information) to gain an exceptional number of skill and tool proficiencies. If done optimally, the Rogue could reasonably get proficiency with every skill and most tools. 

This would likely come down to DM interpretation, but I wanted to see what the community thinks of this. Should Phantom Rogue with Keen Mind be the preeminent skill monkey, or do the rules as written prevent this scheme? 

I think RAW, recalling a memory of being proficient in a given skill (especially when only limited to things "seen" and "heard") would not be enough to maintain proficiency. Proficiency with using a given skill can likely be interpreted as knowing the logic / thought process / feeling behind performing certain tasks, whereas Keen Mind will just let you remember that you did those tasks, not necessarily HOW. 

For example, you use proficiency with Sleight of Hand to pick a lock and then drop that proficiency. You might be able to remember the exact motions you use to pick that lock using Keen Mind, but if you come across a completely different lock you are likely going to need a different approach and you would have lost the mindset behind figuring out what that approach would be when you lost proficiency. 

Its hard for me to put into words exactly, but I think that a skill proficiency represents more of an "active" mindset or way of thinking, whereas Keen Mind only represents individual memories that are not widely applicable to every scenario. 

As with anything it ultimately comes down to how the DM rules it. After all rulings, not rules are ultimately what matter.

However, from a strict RAW reading I do not see this conclusion being supported. The biggest problem comes from the Whispers of the Dead feature itself when it says this: "You lose this proficiency when you use this feature to choose a different proficiency that you lack." This is not your character forgetting a skill, but rather the mechanics of the ability itself telling us you loose the proficiency when you choose a new one.

The second issue I see is that the Keen Mind feat allows you to accurately recall things you have seen or heard. The Whispers of the Dead feature just tells us that "a ghostly presence shares its knowledge with you" without specifying how. So if they share their knowledge with you in a way that is neither seen nor heard then there can be no interaction with the Keen Mind feat. Given that the downtime Training rule tells us it takes 10 work weeks to gain a proficiency I doubt the ghost is sharing its knowledge via ordinary means.

The third issue I see is that a DM could rule that even if the Keen Mind feat lets you recall your interaction with the Ghost that it might not be sufficient to maintain your proficiency.

No. Keen Mind does not grant proficiency.

I wouldn't let it fly, simply on the basis of it not being your mind and knowledge that executes the skill in question.  I see it as a sliver of someone else's mind being accessible to you for a time and when that sliver is gone or replaced, you can't recall it, because YOU never knew it.

Creative idea, however, if flawed, lol.  Too much a game-breaker for any sane DM to permit, honestly.  A PC with ALL proficiencies within a couple weeks of game time?  No chance, lol, those are called Gods in the realms.

I can't imagine actually wanting to upstage the entire party so fiercely that you need to become proficient in all skills. Why bother?

Is this founded on the logic that, because it's called Whispers of the Dead, it is literally a dead person whispering to you how to do or know something?

I would say that the proficiency that comes from whispers of the Dead happen because you are being partially possessed by the ghost of spirit that then acts through you in the moments you need that proficiency. On the other hand the borrowed knowledge spell which gives you proficiency in any skill for an hour requires a material component of a book worth at least 25 gold pieces leaning me to believe that the proficiency comes from instantly downloading the knowledge in the book. I would posit that Keen Mind would work with that thus granting you the ability to have proficiency in all skills but not tools or instruments. As DM I would allow this only because of the opportunity cost of not taking a different Feat/Spell and the gold cost for all the different books you'd need. I'd have them roleplay finding books on the various topics in-game. And ultimately it's not that broken because it's just a slightly better version of the bards jack of all trades ability (wich they get for free) and it's not like they can get expertise in all of the stuff either so no big deal. 🤷🏽‍♂️

I think of it like Dr. Strange in the movie.

He’s a skilled surgeon (whispers of the dead proficiency)

He gets into a car crash and damages his hands (loses said proficiency)

Although he still has the knowledge of his skillful surgical ability his hands just aren’t up to it (keen mind remembers but he still has lost the proficiency)