Classification conjecture for irreducible ordinary \mathcal W_k-modules

Let $k\in\mathbb Z$ with $k\geq -1$, define

$$\mathcal S_k=\left\{(x,y)\in\mathbb C^2\mid h_i(x,y)=0,\;1\leq i\leq k+2\right\},$$

and let $L(x,y)$ denote the irreducible highest-weight $\mathcal W_k$-module with highest weight $(x,y)$. Classification conjecture. The set

$$\left\{L(x,y)\mid (x,y)\in\mathcal S_k\right\}$$

is the set of all irreducible ordinary $\mathcal W_k$-modules. The preceding proposition proves only containment of the equivalence classes of irreducible ordinary modules in this set, so the equality asserted here remains unproved in the supplied text.