The skein lasagna module Poincaré series conjecture for the complex projective plane

Let $P_n(t,q)$ denote the colored Khovanov Poincaré polynomial used in the paper, let $\mathcal{S}_0^2(\overline{\mathbb{CP}^2};h;\mathbb{Q})$ denote the skein lasagna module of $\overline{\mathbb{CP}^2}$ in $H_2$ grading $h$, and let $(a;q)_m$ denote the finite $q$-Pochhammer symbol. Skein lasagna module Poincaré-series conjecture. The Poincaré series in $H_2$ gradings $0$ and $1$ are given by

$$\operatorname{\mathcal{P}}\left(\mathcal{S}_0^2(\overline{\mathbb{CP}^2}; 0; \mathbb{Q}) \right) = \lim_{n \to \infty} q^{-2n}P_{2n}(t,q^{-1})$$ $$= \frac{1}{1+q^{-4}t} \sum_{k = 0}^\infty t^{-2k^2} q^{2k(k+1)} \frac{(-q^{-4}t; q^{2}t^{-2})_{2k+1}}{(q^{2}t^{-2}; q^{2}t^{-2})_{2k}} + t^{-2(k+1)^2} q^{2((k+1)^2 + k)} \frac{(-q^{-4}t; q^{2}t^{-2})_{2k+2}}{(q^{2}t^{-2}; q^{2}t^{-2})_{2k+1}},$$

and

$$\operatorname{\mathcal{P}}\left(\mathcal{S}_0^2(\overline{\mathbb{CP}^2}; 1; \mathbb{Q}) \right) = \lim_{n \to \infty} q^{-2n-1}P_{2n+1}(t,q^{-1})$$ $$= \frac{1}{1+q^{-4}t} \sum_{k = 0}^\infty t^{-2k(k+1)} q^{2(k^2+2k-1)} \frac{(-q^{-4}t; q^{2}t^{-2})_{2k+1}}{(q^{2}t^{-2}; q^{2}t^{-2})_{2k}} + t^{-2k(k+1)} q^{2k(k+2)} \frac{(-q^{-4}t; q^{2}t^{-2})_{2k+2}}{(q^{2}t^{-2}; q^{2}t^{-2})_{2k+1}}.$$

These formulas are conditional in the surrounding discussion on the recursion conjecture and relate the skein lasagna module to limits of colored Khovanov homology. The supplied text does not state that the formulas have been proved or otherwise resolved.