Conjecture on homological dimensions and Lusztig's a-function in category O

Let $\mathfrak{g}$ be a semi-simple complex finite-dimensional Lie algebra, let $W$ be its Weyl group with longest element $w_0$, and let $\mathfrak{t}(w)$ and $\mathfrak{i}(w)$ denote respectively the projective dimensions of the indecomposable tilting and injective modules indexed by $w\in W$. Let $\mathbf{a}:W\to\mathbb{Z}$ be Lusztig's function, constant on two-sided cells. The homological-dimension conjecture. For all $w\in W$,

$$\mathfrak{t}(w)=\mathbf{a}(w),\qquad \mathfrak{i}(w)=2\mathbf{a}(w_0w).$$

The conjecture is motivated by computations in type $A_2$ and other examples, together with the cellwise constancy properties of the relevant functions. The source does not state a resolution.