Sign-reversing involution conjecture for plethysm coefficients

Let $\lambda$, $\mu$, and $\nu$ be partitions, let $l=l(\nu)$, and set

$$A=\{(\pi,T)\mid \pi\in S_l,\ T\in\operatorname{SSTab}(\lambda[\mu];\pi*\nu)\},$$

where $\pi*\nu=(\nu_{\pi(i)}-\pi_i+i)_{1\leq i\leq l}$. Let $\operatorname{SSTab}(\lambda[\mu])$ denote the relevant set of semistandard tableaux, and let $\operatorname{sgn}(\pi)$ be the sign of $\pi$.

Sign-reversing cancellation conjecture. There are a subset $S_0\subset\operatorname{SSTab}(\lambda[\mu])$ and a bijective map

$$\phi\colon A\setminus A_0\longrightarrow A\setminus A_0$$

such that, if $\phi(\pi,T)=(\pi',T')$, then $\operatorname{sgn}(\pi)=-\operatorname{sgn}(\pi')$, where

$$A_0=\{(\pi,T)\in A\mid T\in S_0\}.$$

Such a sign-reversing pairing would cancel all terms outside $A_0$ in the alternating formula $a_{\lambda[\mu]}^{\nu}=\sum_{(\pi,T)\in A}\operatorname{sgn}(\pi)$ and would provide a combinatorial interpretation of the plethysm coefficient. The supplied text presents this as an expected conjecture and gives no evidence of a resolution.