I want to settle the issue with the jump attack once and for all.

Hello, dear developers, administrators, and players of our beloved D&D. I'd like to address a very important and long-standing issue that resonates strongly with a large portion of public opinion.
In the community I'm currently part of, there's a belief that it's impossible to perform an action during a long jump without interrupting the jump. I understand that the rules clearly state that actions can be divided into different acts, but for some reason, a fairly large community on one popular site believes this interpretation requires interrupting the jump.
I'd like clarification and a clear, unambiguous answer for the majority (not a GM's choice, but an answer): is it possible to perform an attack while jumping without interrupting the jump itself?

The PHB and DMG of either edition don't seem to be clear on this, thus it's left to the DM.

I know you don't want a "DM's call" answer, but that's the answer.

The jumping rules are silent on the question. Generalized sense of realism says that you can't do anything complicated, because you are in constant motion while jumping. The rules of drama say you totally can, because it's cool.

Mechanically, one could take the Ready action, saying "I'll attack X when I'm in range", then jump past X, getting a single attack on them.

And I'd be fine with that.

But it's still GM's call.

If the attack is dunking a basketball or shooting a bow, sure.

Anything else, it depends..

I would suggest that it should lie in the DM's level of realistic actions in Combat. Well it is easy to interpret it as a compound move, you must realize that in a jumping attack you are sacrificing solid structure and stance for the minimal gain of momentum. so not only is the character trying to concentrate on placing a lot of detail into movement and increase difficulty with accuracy but he is coming at the creatures from weakened structure and less solid energy transference from an attack made from the ground. 

If the DM is more Rule of Kool. He Slaughters him with advantage from attacking from above 

Attacking during jumping boils down to whether or not jumping is a type of movement that you can break up. I'd argue no because it's not classed as a type of movement—it doesn't use a speed and it has the following clauses:

(High Jump) Either way, each foot of the jump costs a foot of movement

(Long Jump) Either way, each foot you jump costs a foot of movement

Compare this to the language for Climbing

While you’re climbing, each foot of movement costs 1 extra foot

For jumping the syntax is shows a conversion between "jump" and "movement, whereas for climbing (and swimming and crawling too), the syntax just refers to movement.

This shows (IMO) that jumping is not movement (it just costs movement) and thus not being movement it does not qualify under the rules for breaking up your movement.

I've also got to ask:

What are the circumstances where this ever comes up in your group, much less so often that it's a point of contention?

I've had it come up once or twice when a PC wants to attack a flying enemy for example. But it's 100% not a "very important and long-standing issue that resonates strongly with a large portion of public opinion."

As a DM, I absolutely allow something like a single attack or minor interaction during a jump. You can't make multiple attacks in midair that way, but anyone with Extra Attack can use their other attack(s) for the rest of their turn still when I allow this.

But yes, the rules don't specify this. It's the DM's call.