The free-Boson formula for the CKP Darboux transformation

Let $T(q)$ be the CKP Darboux transformation associated with a solution $q$, let $g\in Sp_{\infty}$, and write the corresponding tau function as

$$\tau=g|0\rangle.$$

Let $V=\sum_{l\in \mathbb{Z}+1/2}\mathbb{C}\phi_l$, and choose $\beta\in V$ corresponding to $q$. Free-Boson Darboux transformation formula. Under $T(q)$, the tau function transforms as

$$\tau=g|0\rangle\xrightarrow{T(q)}e^{\frac{\beta^2}{2}}\tau.$$

This gives a free-Boson expression for the action of the CKP Darboux transformation on tau functions. The supplied text does not state whether the formula is established or remains open.