Reciprocity conjecture for recurrence polynomials under Dynkin involution

Assume the $T$-system has tensor-product form $\Lambda'\otimes\widehat{\Lambda}$, where $\Lambda'$ is of type $A_m$, $D_{2m+1}$, or $E_6$, and let $\eta$ be the canonical involution of its Dynkin diagram. For a vertex $v$, let $v'$ have the same $\widehat{\Lambda}$ coordinate and the $\Lambda'$ coordinate obtained from that of $v$ by $\eta$. Let $J_v(z)$ and $J_{v'}(z)$ be the corresponding recurrence polynomials. The reciprocity conjecture. The recurrence polynomials of $v$ and $v'$ have the same coefficients in reverse order; in particular, if $v=v'$, then $J_v(z)$ is palindromic. This generalizes the known symmetry result for the treated family, but the source gives no resolution for the stated generality.