Asymptotic Gross–Kunze inner-product conjecture

Fix $\lambda$, let $W$ be the representation space in the Gross–Kunze construction, and let $\phi$ be the associated map. Let $\langle\cdot,\cdot\rangle_{\mathrm{U}(t)}$ be the unique up to positive scalar inner product on $W$ for which the restriction of the representation to the compact group $\mathrm{U}(t)$ is unitary. Write $|\lambda|$ for the size of $\lambda$.

Asymptotic Gross–Kunze inner-product conjecture. For each $\lambda$ there exists a strictly positive scalar $c$ such that

$$n^{|\lambda|}\langle\phi(u),\phi(v)\rangle=c\langle u,v\rangle_{\mathrm{U}(t)}+O(n^{-1})$$

for all $u,v\in W$.

This conjecture describes the asymptotic behavior of the inner product arising from the Gross–Kunze construction and is intended to support the limiting semidefinite program. The supplied text gives no proof or resolution.