Bilinear equation for the q-deformed Painleve tau function

Let $u,s,q,Z$ be parameters, and let $\mathcal{T}(u,s;q|Z)$ be the function defined from the coefficient function $C(u;q|Z)$ and the $q$-deformed conformal block. The function $C$ may be chosen $u$-inverse invariant, meaning $C(u;q|Z)=C(u^{-1};q|Z)$.

Bilinear equation. The function $\mathcal{T}(u,s;q|Z)$ satisfies

$$Z^{1/4}\mathcal{T}(u,s;q|qZ)\mathcal{T}(u,s;q|q^{-1}Z)=\mathcal{T}(u,s;q|Z)^2+Z^{1/2}\mathcal{T}(uq,s;q|Z)\mathcal{T}(uq^{-1},s;q|Z).$$

This is a bilinear relation underlying the $q$-deformed Painleve tau-function construction. The supplied text gives no separate resolution status.