Projective-dimension conjecture for twisted tilting modules

Let $W$ be the Weyl group with longest element $w_0$, let $\theta_x$ be the projective functor indexed by $x\in W$, and let $\Delta_z$ and $\nabla_z$ denote the standard and costandard modules indexed by $z\in W$. Write $\mathbf{a}$ for the relevant Kazhdan–Lusztig $a$-function and $\mathrm{proj.dim}$ for projective dimension. Projective-dimension conjecture. For $x,y\in W$, one has

$$\mathrm{proj.dim}\, \theta_x\nabla_y=\mathbf{a}(w_0 x)+ \mathrm{proj.dim}\,\theta_x\Delta_{w_0y}.$$

The claim is proposed as a connection between the projective dimensions of twisted tilting and twisted standard modules. Several boundary cases are listed in the source, but the general equality remains open there.