Plancherel formula and Lusztig's a-function conjecture

Let $\Pi$ be a class of representations appearing in the Plancherel theorem, and let $\Omega$ be the map from such classes to two-sided cells; write $\Gamma=\Omega(\Pi)$. Let $\mathbf{a}$ denote Lusztig's $\mathbf{a}$-function and let $\nu_{\mathsf{q}}(\Pi)$ be the invariant associated with $\Pi$ in the paper. Plancherel a-function conjecture. For every $w\in\Gamma$, one has

$$\mathbf{a}(w)=\frac{\nu_{\mathsf{q}}(\Pi)}{2}.$$

This conjecture would relate the $\mathbf{a}$-function, which is constant on the relevant two-sided cell, to the Plancherel-theoretic invariant of the representation class.