Affine Gaudin Bethe ansatz conjecture for qKdV non-local integrals of motion

Let $\boldsymbol G_n$ be the non-local integrals of motion of the quantum KdV model acting on a Virasoro Verma module, and let the subspace of level $N$ be parametrized by two sets of Bethe roots

$$\bar s=(\bar s_i)_{i=1,\dots,N},\qquad \bar t=(\bar t_j)_{j=1,\dots,N}.$$

Let $r_n$ be supersymmetric polynomials in infinitely many variables, with degree $n$.

Affine Gaudin Bethe ansatz conjecture. Every eigenvalue of $\boldsymbol G_n$ in the level-$N$ subspace has the form $r_n(\bar s,\bar t)$ for a solution of the affine Gaudin Bethe ansatz equations with $N_0=N_1=N$. Moreover,

$$r_1(\bar s,\bar t)=\boldsymbol G_1^{(\mathrm{vac})}\left(1-\frac{1}{v}\frac{2(1-2\beta)}{1-\beta+2P}\sum_{i=1}^{N}(\bar s_i-\bar t_i)\right),$$

and, for generic $r$ and $\hat\pi$, the number of such solutions equals the number of partitions of $N$.

This conjecture would identify the qKdV non-local spectrum with affine Gaudin Bethe roots and match the expected partition count of level-$N$ states. It is presented as an expectation depending on the preceding quantum-toroidal conjecture, and no proof is supplied in the cited passage.