Classification conjecture for K-spherical representations of infinite-dimensional general linear groups

Let $\mathbf G=\mathrm{GL}(k\infty,\mathbb{Q}_p)$, and let $\mathbf K$ be its compact subgroup. For each positive integer $l$ and each $r\in\mathrm{Sp}(2kl,\mathbb{Q}_p)$, let $\rho_r$ be the representation obtained from the embedding into $\mathrm{Sp}(2l\alpha+2lk\infty,\mathbb{Q}_p)$ described above after setting $\alpha=0$. Let $\varphi:\mathbb{Q}_p^\times\to\mathbb{C}^\times$ be a character. Classification conjecture. Any $\mathbf K$-spherical representation of $\mathrm{GL}(k\infty,\mathbb{Q}_p)$ is a subrepresentation of $\varphi(\det(g))\,\rho_r(g)$, where the representations $\rho_r$ are parametrized by

$$\bigcup_l \mathrm{GL}(l,\mathbb{Q}_p)\setminus \mathrm{Sp}(2kl,\mathbb{Q}_p)/\mathrm{Sp}(2kl,\mathbb{O}_p).$$

This proposes a parametrization of all spherical representations by the displayed double-coset data, up to twisting by characters of $\mathbb{Q}_p^\times$. The parser provides no evidence that the statement has been proved or disproved.