Endomorphism-algebra conjecture for objects in the diagrammatic q-Schur category

Let $\lambda\rightarrow\ru(n,d)$ be arbitrary, and let $mu\rightarrow\ru(n,d)$ be obtained from $\lambda$ by moving all zero entries to the end while preserving the relative order of the nonzero entries; for example, $(2,0,1)$ gives $(2,1,0)$. Write $S\banPi_{mu}$ for the algebra appearing in the source. Endomorphism-algebra conjecture.

$$\operatorname{End}_{\mathcal{S}(n,d)}(1_{\lambda})\cong S\Pi_{\mu}.$$

The conjecture predicts an explicit description of every such endomorphism algebra and is motivated by relations among bubbles of adjacent colors and equivalences in the Karoubi envelope. The source gives no resolution.