The W-algebra conjecture for the generalized Witten potential

Let $Z(\mathbf{t})$ be the exponential of the $\mathrm{KdV}_r$ potential function. Let $W_r^+$ denote the relevant algebra of differential operators, containing the Virasoro generators $\{L_n\}_{n\geq -1}$. W-algebra conjecture. There exists a collection of differential operators forming a $W_r^+$ algebra, with the Virasoro generators $\{L_n\}_{n\geq -1}$ as a subset, such that this collection annihilates and completely determines $Z(\mathbf{t})$. This is presented as an alternative formulation of the generalized Witten conjecture; the source gives no resolution of the claim.