The even-spin W-infinity extension conjecture for the symplectic universal algebra

Let $\mathcal{W}^{\mathfrak{sp}}_{\infty}(c,\mathsf{k})$ be the universal two-parameter vertex algebra, let $\mathrm{H}^0_{\ydiagram{2}}$ denote its Virasoro-type reduction, and let $R$ be the original base ring with extension $\widehat{R}=R(\gamma(c,\mathsf{k}))$. The parameters $c_1$ and $c_2$ are the central charges defined in the source. The even-spin W-infinity extension conjecture. The reduced algebra is an extension of the tensor product

$$\mathcal{W}^{\mathrm{ev}}_{\infty}(c_1,\lambda_1)\underset{\widehat{R}}{\otimes}\mathcal{W}^{\mathrm{ev}}_{\infty}(c_2,\lambda_2)\hookrightarrow \widehat{R}\underset{R}{\otimes}\mathrm{H}^0_{\ydiagram{2}}(\mathcal{W}^{\mathfrak{sp}}_{\infty}(c,\mathsf{k})).$$

This is motivated by the two commuting Virasoro vectors and associated primary weight-four vectors constructed after base change; the extension statement is supported by explicit low-weight OPE calculations but remains conjectural.