I ride a belt drive bike with a Shimano Nexus SG-C6001-8D IGH. I'm aware of the normal cable adjustment procedure (aligning yellow marks on the hub), I've done this.
When I ride normally, shifting is instantaneous and smooth. However, when I move off at lights, I usually use gear 3, and I've noticed that if I accelerate hard, the hub will not shift into gear 4 after I change it; instead, it only shifts after I go up to 5 (and then it immediately goes 3 -> 5, so gear 4 is effectively skipped over). Sometimes it also happens that if I shift 3->4 and leave it there, without going to 5, 4 will eventually engage after 5 seconds or so.
I should point out I don't attempt to shift under load; I accelerate in gear 3, release pressure on the pedals, then shift and continue.
As said above, this 3->4->5 shifting issue only happens when accelerating. If I ride normally at a constant speed, those gears have no issues.
I know for a fact the shift cable needs to be cleaned, there's likely water inside, as when it was -5°C/23°F outside once, the cable froze in its housing. Could this issue be caused by the cable, or is it a hub problem? I suspect the cable but I'm not sure why everything would be fine when not applying load. For reference, I'd estimate the hub has at most 5000km/3000 miles on it.