The image-equals-projective conjecture for Kostant's problem

Let $W$ be the Weyl group, let $\boldsymbol{R}$ be a right cell, and let $\widehat{\boldsymbol{R}}=\{x\in W:x\leq_{\mathtt{R}}w\text{ for some }w\in\boldsymbol{R}\}$. Let $\mathcal{O}_0^{\widehat{\boldsymbol{R}}}$ be the full subcategory of $\mathcal{O}_0$ whose simple objects are $L(x)$ for $x\in\widehat{\boldsymbol{R}}$, closed under isomorphisms and extensions. Write $P^{\widehat{\boldsymbol{R}}}(w)=\mathrm{Z}^{\widehat{\boldsymbol{R}}}_0P(w)$ for $w\in\widehat{\boldsymbol{R}}$. If $\mathbf{w}\in\boldsymbol{R}$ is the unique involution in $\boldsymbol{R}$, let $D^{\widehat{\boldsymbol{R}}}$ be the image of the unique up to scalar nonzero homomorphism $P^{\widehat{\boldsymbol{R}}}(e)\rightarrow P^{\widehat{\boldsymbol{R}}}(\mathbf{w})$. Image-equals-projective conjecture. One has

$$D^{\widehat{\boldsymbol{R}}}=P^{\widehat{\boldsymbol{R}}}(e).$$

This assertion concerns the structure of projective modules in truncated principal blocks of category $\mathcal{O}$ and is formulated in the context of Kostant's problem. Its resolution status is not specified in the supplied source material.