Dominant-dimension inequality for tensor powers and Schur algebras

Let $K$ be a field, let $S_R(n,d)$ denote the Schur algebra over a commutative ring $R$, and let $V$ be its natural module. For $n,d\in\mathbb{N}$, write $V^{\otimes d}-\operatorname{domdim}(S_R(n,d),R)$ for the relative dominant dimension associated with $V^{\otimes d}$. Tensor-power dominant-dimension conjecture. For all $n,d\in\mathbb{N}$ and every commutative ring $R$,

$$V^{\otimes d}-\operatorname{domdim}(S_R(n,d),R)\geq \operatorname{domdim}(S_R(n,d),R).$$

The statement is motivated by the preceding discussion of partial tilting modules and projective-injective modules for Schur algebras. The supplied text gives no evidence that the inequality has been proved or disproved.