Coset conjecture for symplectic bosons and a subregular W-algebra

For $n\geq 2$, let $\mathcal{S}(n(n+1))$ denote the symplectic-boson vertex algebra on a $2n(n+1)$-dimensional symplectic vector space, with its natural $\mathfrak{sl}_n[t]$ action. Let $f_{n-1,n+1}$ be the nilpotent element specified by the corresponding partition of $2n$.

Coset conjecture. For all $n\geq 2$, there is an isomorphism

$$\mathcal{S}(n(n+1))^{\mathfrak{sl}_n[t]} \cong \mathcal{W}_{2-2n}(\mathfrak{sl}_{2n},f_{n-1,n+1}).$$

The conjecture identifies a current-algebra invariant subalgebra of a symplectic-boson vertex algebra with a simple $\mathcal{W}$-algebra at level $2-2n$. The supplied source gives no resolution status beyond presenting the assertion, so its general case remains open here.