Koszulity conjecture for the degree-zero and degree-one subalgebra of the extension algebra

Let $A$ be the algebra governing the principal block of category $\mathcal{O}$, let $\Delta$ be the direct sum of the relevant Verma modules, and let $\mathtt{E}$ be the graded extension algebra under consideration. Denote its degree-$i$ component by $\mathtt{E}_i$. Koszulity conjecture. The subalgebra of $\mathtt{E}$ generated by $\mathtt{E}_0$ and $\mathtt{E}_1$ is Koszul.

This is motivated by the Koszulity of $\operatorname{End}^*_A(\Delta)$ and by known Koszulity results in multiplicity-free cases. The source does not provide a resolution of the conjecture.