Hypergeometric BKP conjecture for the Kontsevich–Witten tau-function

Let $\operatorname{DP}$ be the set of strict partitions, and let a hypergeometric BKP tau-function be expressed as

$$\tau=\sum_{\lambda\in\operatorname{DP}}r_\lambda Q_\lambda({\bf t}/2)Q_\lambda({\bf t}^*/2),$$

with $r_\lambda=e^{\sum_{j=1}^{\ell(\lambda)}\xi(\lambda_j)}$. For $k\geq 1$, define

$$A_k=\prod_{j=1}^k\frac{(6j-1)(6j-5)}{16},$$

and, for $\beta\neq0$, define $\xi_{KW}$ by

$$e^{\xi_{KW}(3k)}=\hbar^kA_k,$$ $$e^{\xi_{KW}(3k-1)}=-\hbar^{k-1/3}\frac{2}{(6k-1)\beta}A_k,$$ $$e^{\xi_{KW}(3k-2)}=\hbar^{k-2/3}\frac{8\beta}{6k-1}A_k.$$

Set $r^{KW}_\lambda=e^{\sum_{j=1}^{\ell(\lambda)}\xi_{KW}(\lambda_j)}$. Kontsevich–Witten hypergeometric BKP conjecture. The properly normalized Kontsevich–Witten tau-function $\tau_{KW}({\bf t}/2)$ is a hypergeometric tau-function of the BKP hierarchy with these coefficients and with $t_k^*=\frac{2}{3}\delta_{k,3}$. The claim is presented as implied by the conjectural Schur Q-function expansion and is not resolved in the supplied text.