Affine evacuation weight-vector conjecture

Let $w$ be an affine permutation with AMBC data

$$w \overset{\operatorname{AMBC}}{\longleftrightarrow}(P,Q,\rho),$$

where $P$ and $Q$ are tabloids of shape $\lambda$, and let $r(w)$ denote the rotation of $w$. For a vector, call its dominant representative the representative specified in the cited AMBC conventions. Affine evacuation weight-vector conjecture. There exists a function $d$ from tabloids of shape $\lambda$ to integer vectors of length $\ell(\lambda)$ such that, for every $w$, $\rho(r(w))$ is the dominant representative of

$$-\rho(w)+d(Q(w))-d(P(w)).$$

The claim is motivated by the observation that rotation moves entries across the diagonal in a way suggesting $\rho'=-\rho$, but the existence of the correction function $d$ and the stated formula remain conjectural.