Kåhrström's multiplicity-one conjecture for Duflo translates

Let $\mathfrak{g}$ be a semisimple complex Lie algebra with Weyl group $W$, let $d,y\in W$ with $d$ a Duflo element, and let $\theta_d$ be the corresponding indecomposable projective functor. Let $M$ be an indecomposable direct summand of $\theta_dL_y$, let $\mathcal{O}_0^{\mathbb{Z}}$ be the graded principal block, and let $\langle \mathbf{a}(d)\rangle$ denote the grading shift by Lusztig's $\mathbf{a}$-function. Kåhrström's conjecture. For every such indecomposable summand $M$,

$$\dim\operatorname{hom}\bigl(M,L_y\langle\mathbf{a}(d)\rangle\bigr)=1.$$

The conjecture is motivated by the study of graded simple tops of translated simple modules and follows a theorem giving a lower bound on the grading degree in which the relevant homomorphisms can occur. The supplied text gives no resolution status.