Lusztig's symmetry conjecture for dimension polynomials

Let $W$ be a Weyl group, let $\mathcal{J}$ be its set of near involutions, and write $\dim(M_w)=P_w(q)$, where $P_w(t)\in\mathbb{Q}[t]$. For $w\in\mathcal{J}$, let $\mathcal{L}_D(w)\subset S$ be its left descent set, and let $\nu$ be the exponent appearing in the conjectured symmetry. Lusztig's symmetry conjecture. There exists an involution $w\leftrightarrow\widetilde{w}$ of $\mathcal{J}$ such that

$$t^\nu P_w(1/t)=\pm P_{\widetilde{w}}(t),\qquad \mathcal{L}_D(\widetilde{w})=S\setminus\mathcal{L}_D(w).$$

The paper states that this conjecture is false for the characters $M_w$ constructed there, so it is refuted in this setting.