Douglas's strengthened essential normality conjecture

Let $K$ be a finite dimensional Hilbert space, let $H^2_d$ denote Drury–Arveson space, and let $L\subseteq H^2_d\otimes K$ be a graded submodule. Write $\dim(L)$ for the dimension parameter used in the source. Douglas's conjecture. The quotient $(H^2_d\otimes K)/L$ is $p$-essentially normal for all $p>\dim(L)$. This strengthens the bound $p>d$ in Arveson's conjecture. It was verified in the classes of submodules where Arveson's conjecture was known, but remains open in general.