Asymptotic Clifford–symplectic Howe-type duality

Fix $t$ and let $n$ be sufficiently large. Let $C_{\mathbf{1}_t}$ be the code space appearing in the construction, let $T'$ be the associated symplectic space, let $\mathrm{Sp}(T')$ be its symplectic group, let $\mathrm{O}(V)$ be the orthogonal group acting on $V$, and let $\mathrm{RCl}$ be the real Clifford group. Write $\mathrm{Irr}(G)$ for the set of irreducible representations of a group $G$. Asymptotic Clifford–symplectic duality. There exists a subspace $\mathcal{L}\subset C_{\mathbf{1}_t}$ such that

$$\frac{\mathrm{dim}\, C_{\mathbf{1}_t}-\mathrm{dim}\,\mathcal{L}}{\mathrm{dim}\, C_{\mathbf{1}_t}}=o(\exp(-n)),$$

and an injective function

$$\theta:\mathrm{Irr}\,\mathrm{Sp}(T')\to\mathrm{Irr}\,\mathrm{O}(V)\subset\mathrm{Irr}\,\mathrm{RCl}$$

such that, as an $\mathrm{Sp}(T')\times\mathrm{RCl}$-representation,

$$\mathcal{L}\simeq\bigoplus_{\tau\in\mathrm{Irr}\,\mathrm{Sp}(T')}\tau\otimes\theta(\tau).$$

This asserts an asymptotically full subspace carrying a multiplicity-free duality between the symplectic and real Clifford representation theories. The surrounding discussion motivates it by the asymptotic count of orthogonal-group orbits, but the supplied text gives no resolution evidence, so it remains open.