Conjectural algebraic form of the torus one-point block recursion

Let $K_N(\Delta,\Delta_1,\beta)$ be defined by the recursion in the source, and let $P_N$ be the quantity occurring there, $Q_N:=P_N-N$, and $B_s$ the Laurent-polynomial basis defined in the source. A Laurent-polynomial is $b$-symmetric when it is invariant under the relevant $b$-symmetry transformation. Algebraic-form conjecture. $K_N$ is a $b$-symmetric Laurent-polynomial in $\beta$ of degree $Q_N$, with coefficients polynomial in $\Delta$ and $\Delta_1$. More precisely, it has the form

$$K_N(\Delta,\Delta_1,\beta)=\biggl\{\sum_{i=0}^{P_N-2}\Delta^{P_N-1-i}\biggl(\sum_{s=0}^{\min(Q_N,i)}\sum_{j=0}^{\min(2N-2,2i-2s)}(-1)^jC^N_{ijs}\Delta_1^jB_s\biggr)\biggr\}+\Delta_1(\Delta_1-1)\sum_{s=0}^{Q_N}\sum_{j=0}^{2N-4}(-1)^jC^N_{(P_N-1)js}\Delta_1^jB_s,$$

where the coefficients satisfy $C^N_{ijs}\in\mathbb{Q}_+$. This form is inferred from the first four calculated orders and remains unproved in general.