The c=1 and c=-2 tau-function correspondence conjecture

Let $u=q^{2\sigma}$ and let $\tau$ and $\uptau$ denote the $c=1$ and $c=-2$ Painlevé VI tau functions, respectively. Let $\td{u}=\td{u}(u)$ and $\td{s}=\td{s}(s,\sigma)$ be functions, and let $f(u;q)$ and $h_k(u;q)$, $k=0,t,1,\infty$, be functions that are $q$-periodic in $z$. Tau-function correspondence conjecture. There exist such functions satisfying the four displayed identities relating the $c=1$ and $c=-2$ tau functions. The conjecture is motivated by the expected general-solution interpretation of the preceding $q$-Painlevé VI tau-function conjecture; the supplied text gives no resolution status.