Recursive Poincaré-polynomial conjecture for Khovanov homology of $T(n,n)$

Let $K_n(t,q)$ denote the Poincaré polynomial of the Khovanov homology of $T(n+1,n)$, and let $L_n(t,q)$ denote that of $T(n,n)$. Let $C_n=\frac{1}{n+1}\binom{2n}{n}$ be the $n$th Catalan number. Recursive Poincaré-polynomial conjecture. The Poincaré polynomial $L_n$ is recursively defined by

$$L_0=1,\qquad L_1=q^{-1}+q,$$

and, for $n\ge2$,

$$\begin{aligned} L_n={}&t^{2n-2}(q^{6n-8}+q^{6n-6})L_{n-2}+\sum_{i=1}^{\lfloor (n-1)/2\rfloor}C_{i-1}t^{2i(n-i)}q^{6i(n-i)}L_{n-2i}\\ &+\sum_{i=0}^{\lfloor (n-2)/2\rfloor}\left(\binom{n-2}{i}-\binom{n-2}{i-1}\right)t^{2i(n-i)}q^{6i(n-i)+n-2i-1}K_{n-2i-1}. \end{aligned}$$

This recurrence is presented as equivalent to the saddle-surjectivity conjecture and would determine the rational Khovanov homology of the torus links $T(n,n)$ from the corresponding torus-knot polynomials.