Back heel up or down in gyaku Zuki (reverse punch) at kime? (in kumite)

I am being asked that question by many people so here is the point of view of budo karate, Nishiyama system.

The answer is not so simple and requires understanding the priniples of stance and delivery of force. Basically the answer is heel down, heel firm on the floor, but this is the ideal, the kata and basics show us the optimal.
In basics (kihon) and kata there is no compromise, the heel should strictly be firmly down (unless one is injured), and if one has stiff ankles, the stance should adjusted, shortened so the back heel and outer edge of the foot are firm on the floor.
Now, in kumite (applications), one can violate the form as long as the underlying principles of the kata are not violated.

Lets get into it:
Heel on the floor allows for stronger driving from the back leg and supporting the technique at kime, engaging the back leg fully, back heel down make stronger connection of the back leg to body center, it allows at kime for breath pressure to front foot and reaction of this pressure to back foot, which makes for a cycle of energy, unbroken wheel of energy if you will. This allows for more power transmission, and keeping potential energy, and therefore smooth transition without gaps and initiating even the smallest action from the feet and ground reaction.
Don’t forget that kime should also be best condition to start next technique.
When the back foot is firm on the floor, at kime when we make pressure to floor, there will be no escape, dissipation of energy, and the reaction from this pressure will be absorbed through the kinetic chain and fully delivered to line of technique, whilst if the heel is up some of the pressure will escape.
This might not be important for a boxer or kick boxer since they don’t use the concept of pressure to floor at kime but rely on momentum.
Maybe not even for a sport karate practitioner where the fist or foot reaching the target is considered a scoring point and Todome is not the goal.
When someone throws a baseball, the heel coming off the ground is necessary since it is additional power of the ankle, but there is not reaction force at contact.
For us in karate, pressure is main energy; especially in short space where the luxury of big momentum is not available, and we need to avoid recoil, bouncing at impact.
Sensei Nishiyama had us punching the pads and at impact confirming the back foot receiving pressure back, before transition to kamae.

In reality, sensei Nishiyama was the only person I knew who could keep his foot always firm on the floor even when while shifting. I try very hard, but I admit that sometimes my heel still comes off the floor.

Here is the answer:
We should strive to keep as close to the optimal as possible, but what really important is that we don’t lift the heel so much to the point that the weight and body center shift over to the front foot, since that will make us floating, we will lose the interaction between the feet, and between the feet and the body center, we will lose potential energy and the ability to make pressure to floor.
So even if the back heel comes off the floor slightly, one must make sure to keep the back leg engaged, keep the cycle of energy, pressure from back foot through center to front foot and reaction to back foot which in turn delivers the energy to line of technique. Keep the ability to make pressure to floor with least leaking of energy.
One should make sure to avoid pronation of the back foot (rolling toward the inner edge and lifting the outer edge), avoid over stretch of the back leg, since it will become like an anchor that contradict energy direction rather than support it.
Those faults can also create injury over time.
Keep optimal space between the feet.

Similarly, when talking about posture in basics we want to be upright as possible, but in reality we are not mummies, and sometimes applications require that we lean in certain way, yet, we must keep as close to optimal posture as possible, and even when leaning, we want to keep neutral spine as possible, connection through the kinetic chain, and optimal length for function of all torso musculature.
So once more, the form can be violated as long as its underlying principles are kept.