The eventual parity conjecture for positions with many largest piles

Let $n\geq 3$, and let $a_i$ denote the number of piles of size $i$ for $1\leq i\leq n$. Assume that

$$a_n\geq 2n-4.$$

Eventual parity conjecture. The outcome class of the position is determined solely by the parities of $a_1,a_2,\ldots,a_{n-1}$. The source reports preliminary calculations for small pile sizes and presents this as a general pattern; no proof or resolution is given.