Dominant Schur module conjecture for Veronese syzygies

Let $K_{p,q}(\mathbb P^n,b;d)$ be the Koszul cohomology group of the $d$-uple Veronese embedding, let $\mathrm E_{p,q}(\mathbb P^n,b;d)$ be its monomial syzygy construction, and let $\operatorname{domWeights}$ denote the set of most dominant Schur-module weights occurring in a representation. Dominant Schur module conjecture. For all $n,d,b,p$ and $q$,

$$\operatorname{domWeights}\,\mathrm E_{p,q}(\mathbb P^n,b;d)=\operatorname{domWeights}\,K_{p,q}(\mathbb P^n,b;d).$$

Thus the monomial syzygies are conjectured not only to control vanishing and nonvanishing, but also to determine the most dominant Schur-module weights. The authors report computational confirmations, including cases with $d=7$, but the assertion is not proved in general.