Heisenberg–Virasoro extension relating the KW and Hodge tau-functions

Let $\tau_{KW}({\bf t^o})$ and $\tau_{Hodge}({\bf t},u)$ denote the Kontsevich–Witten and Hodge tau-functions, respectively. Let $\beta=u^3$, let $\widehat{L}_0$ be the grading operator, and let $\widehat{L}_+$ be the Virasoro operator defined by the series $f_+$ from the cited equation. Define

$$\widehat{G}_+=\beta^{-\frac{4}{3}\widehat{L}_0}e^{\widehat{L}_+}\beta^{\frac{4}{3}\widehat{L}_0}\in {\bf Vir}_+.$$

Heisenberg–Virasoro extension. The KW and Hodge tau-functions are related by

$$\tau_{KW}({\bf t^o})=\widehat{G}_+\,\tau_{Hodge}({\bf t},u).$$

This proposed extension incorporates the KW tau-function into the previously known relation between the Hodge and Hurwitz tau-functions via the Heisenberg–Virasoro group. The supplied text does not establish the relation or indicate whether it has since been proved or disproved.