Bilinear tau-function conjecture for q-Painlevé V

Let $\tau_i$ $(i=1,\dots,6)$ be the tau functions associated with the $q$-Painlevé V equation, with parameters $\theta_*$, $\theta_t$, and $\theta_0$. Write $\overline{f}(t)=f(qt)$ and $\underline{f}(t)=f(t/q)$. Bilinear tau-function conjecture. The tau functions satisfy

$$\begin{gathered} \tau_1\tau_2-q^{-\theta_*}(q-1)^{1/2}t\tau_3\tau_4-(1-q^{-\theta_*}t)\overline{\tau_1}\underline{\tau_2}=0,\\ (q-1)^{-1/2}\tau_1\tau_2-\tau_3\tau_4+(1-q^{-\theta_*}t)q^{2\theta_t}\underline{\tau_5}\overline{\tau_6}=0,\\ (q-1)^{-1/2}\tau_1\tau_2-q^{2\theta_t}\tau_3\tau_4+q^{2\theta_t}\tau_5\tau_6=0,\\ \tau_1\underline{\tau_2}+q^{\theta_t-1/2}(q-1)^{1/2}t\underline{\tau_5}\tau_6-\underline{\tau_1}\tau_2=0,\\ (q-1)^{-1/2}\tau_1\underline{\tau_2}+q^{\theta_0+2\theta_t}\underline{\tau_5}\tau_6-q^{\theta_t}\underline{\tau_3}\tau_4=0,\\ (q-1)^{-1/2}\tau_1\underline{\tau_2}+q^{-\theta_0+2\theta_t}\underline{\tau_5}\tau_6-q^{\theta_t}\tau_3\underline{\tau_4}=0. \end{gathered}$$

Consequently, the functions

$$y=q^{-\theta_*-1}(q-1)^{1/2}t\frac{\tau_3\tau_4}{\tau_1\tau_2},\qquad z=-q^{\theta_t-\theta_*/2-1}(q-1)^{1/2}t\frac{\underline{\tau_5}\tau_6}{\tau_1\underline{\tau_2}}$$

solve the $q$-Painlevé V equation. This conjecture arises from degenerating the corresponding $q$-Painlevé VI tau-function construction; the source supplies no evidence that it has been proved or disproved.