Sibirjak, there was a referendum in Subcarpathian Rus (now Zakarpattje) I think in 1991. People voted for autonomy (I think 72% or 78% or so were for autonomy). Ukrainian government ignored it. Well, in the context of political intimidation of Subcarpathian and Rusyn activists, ukrainization, and overall political situation in the country, demanding independence would be perfectly justified by now.
But they do not do it (yet), because their (underground) leaders know that the region is not prepared for that. Does it have capacities needed for a functioning state? Would there be a stable and relatively unified political platform that would ensure stability needed for economic growth? Would neighbouring countries accept their independence? Who would protect them from retaliation of Kiev? Russia? But being tied to Russia is really no win.
So no independence at the moment.

And Kiev ignores them practically because heeding their demands could be a catalyst for disintegration of Ukraine.

What about joining Slovakia. I think thats a viable option.