Eventual tail conjecture for four-stack Twyst-off positions

Consider positions in Twyst-off with four stacks, written $(a,b,c,a)$, and let a $P$ position be one from which the next player has no winning move.

Eventual tail conjecture. For every $a>0$, there exist integers $c_0>b_0>a$ such that

$$(a,b_0,c,a)\text{ is a }P\text{ position if and only if }c\geq c_0.$$

This predicts an eventual interval of $P$ positions as the third stack grows, for a suitable fixed second stack. The statement appears after observations that remaining four-stack $P$ positions are complicated but exhibit long-run patterns; no proof or resolution is supplied.