Recursive formula conjecture for the deformed Gaiotto-state inner product

Let $q$ and $t$ be generic complex numbers, let $\Lambda$ be a non-zero complex number, and let $|G\rangle$ be the deformed Gaiotto state of the deformed Virasoro algebra. Introduce $Q$ by

$$h=Q^{1/2}+Q^{-1/2}.$$

Write its inner product as

$$\langle G|G\rangle=F(\Lambda,Q,q,t)=\sum_{n=0}^{\infty}(\Lambda^4t/q)^nF_n(Q,q,t),$$

where each $F_n(Q,q,t)$ is rational in $Q,q,t$. For integers $r,s$ with $1\leq rs\leq n$, define

$$G(r,s;q,t)=-\operatorname{sgn}(r)q^rt^{-s}\prod_{\substack{-|r|\leq i\leq |r|-1,\\- |s|\leq j\leq |s|-1,\\(i,j)\neq(0,0)}}\frac{1}{1-q^it^{-j}},$$

where $\operatorname{sgn}(r)=1$ for $r>0$ and $\operatorname{sgn}(r)=-1$ for $r<0$.

Recursive formula conjecture. The coefficients satisfy

$$F_n(Q,q,t)=\delta_{n,0}+\sum_{\substack{r,s\in\mathbb{Z}\\1\leq rs\leq n}}\frac{G(r,s;q,t)F_{n-rs}(q^rt^s,q,t)}{Q-q^rt^{-s}}.$$

The conjecture gives an explicit recursive description of the deformed Gaiotto-state inner product, which is the quantity appearing in the five-dimensional AGT correspondence. The paper proves that the five-dimensional pure SU(2) Nekrasov partition function satisfies the same recursion, thereby reducing the AGT conjecture to this proposed formula; the recursive formula itself is not proved in the stated result.