CFT-type Poisson vertex algebra classification conjecture

Let $A=\mathbb{C}[D]L\oplus\mathbb{C}[D]W_1\oplus\cdots\oplus\mathbb{C}[D]W_{\ell-1}\oplus\mathbb{C}C$ be a $\mathbb{C}[D]$-module with $DC=0$. Endow $S(A)$ with a Poisson $\lambda$-bracket such that $\{L_\lambda L\}=(D+2\lambda)L+\lambda^3C$ and each $W_j$ is primary of conformal weight $\Delta_j\in\mathbb{N}$. Assume that $A$ has no proper nonzero $\mathbb{C}[D]$-submodule $I$ for which $IS(A)$ is a PVA ideal. CFT-type PVA classification conjecture. The quotient $S(A)/(C-c)$ is isomorphic to a classical $W$-algebra $\mathcal{W}^k(\mathfrak{g},f)$, where $\mathfrak{g}$ is a simple Lie algebra, including the one-dimensional case, $f$ is a nilpotent element of $\mathfrak{g}$, and $c=-k(x|x)$ for an $sl_2$-triple containing $f,x$ with $[x,f]=-f$. The preceding rank-two and Neveu--Schwarz cases support this proposed classification, but the general statement is unproved.