Symmetry conjecture for generating functions of even and odd 3412-avoiding involutions

Let $I(3412,[l_1,\ldots,l_m])$ denote the involutions under consideration, and let $F^-_{[l_1,\ldots,l_m]}(x)$ and $F^+_{[l_1,\ldots,l_m]}(x)$ be the generating functions for the odd and even involutions, respectively.

Symmetry conjecture. For all $m \ge 1$ and all $l_1, \ldots, l_m \ge 1$, the generating functions for the even involutions in $I(3412, [l_1, \ldots, l_m])$ and for the odd involutions in $I(3412, [l_1, \ldots, l_m])$ are symmetric in $l_1, \ldots, l_m$.

The conjecture combines the separately stated symmetry claims for $F^-_{[l_1,\ldots,l_m]}(x)$ and $F^+_{[l_1,\ldots,l_m]}(x)$. The negative case was verified computationally for $m=3$ with $l_i\le 24$ and for $m=4$ with $l_i\le 20$; the general statement remains open.