Type C coset conjecture for principal minimal W-algebras

Let $n\geq 2$ and let $k$ satisfy that $k+1/2$ is a positive integer. Define

$${\mathcal C}_k(n)=\operatorname{Com}(L_{k+1/2}({\mathfrak {sp}}_{2n-2}),{\mathcal W}_k({\mathfrak {sp}}_{2n},e_{-\theta})).$$

Set $m=k+1/2$ and choose $s$ so that

$$s+(k+3/2)=\frac{n+k+1/2}{2n+2k+2}.$$

Type C coset conjecture. The vertex algebra ${\mathcal C}_k(n)$ is isomorphic to the $C_2$-cofinite, rational principal ${\mathcal W}$-algebra ${\mathcal W}_s({\mathfrak {sp}}_{2m},f_{\operatorname{prin}})$.

The central charges agree, and the target is rational and $C_2$-cofinite because $s$ is a nondegenerate admissible level. The conjecture is known for $n=2$ and $k=1/2,3/2$, while the general case remains open.