I think the KHSAA will step in to help the two schools get the game played. It's just a disagreement about when — not whether — the game should be made up, and nothing more than that, and that's why they'll have to step in.
Lincoln's athletic staff seems to be making a good-faith effort to try to get in a game that they weren't a part of postponing, but Pulaski doesn't want to be inconvenienced if possible as it thinks about its playoff future. The onus is on Lincoln to try reschedule the game, and it sounds like they're trying pretty hard, but I can absolutely understand Pulaski's point as well. That's why the KHSAA will have to mediate, and ultimately there's going to be a solution that one side isn't going to like.
I think that side will be Pulaski, because they're going to be told that Lincoln deserves the chance to play. It's not like a normal forfeit situation in which one team just doesn't want to play the other. Lincoln wants to play, and in fact needs to play to be eligible for the playoffs. So my guess is one way or another, it's going to get played.