Determinant formula for sinh-Gordon one-point functions in the fermionic basis

Let $I^+$, $\overline I^+$, $I^-$, and $\overline I^-$ be the index sets specifying fermionic descendants, let $\Phi_\alpha(0)$ be the primary field of parameter $\alpha$, and let $\Theta^{\textrm{shG}}_R$ be the sinh-Gordon dressed function defined by the preceding integral equations. For two sets $A=\{a_j\}_{j=1}^n$ and $B=\{b_j\}_{j=1}^n$, define

$$\mathcal D(A\vert B\vert \alpha) \doteq \left(\prod_{\ell=1}^n \frac{\operatorname{sgn}(a_\ell)\operatorname{sgn}(b_\ell)}{\pi} \right) \det \left[\Theta^{\textrm{shG}}_R(ia_j,ib_k\vert \alpha)-\pi \delta_{a_j,-b_k} \operatorname{sgn}(a_j) t_{a_j}(\alpha)\right]_{j,k=1}^n.$$

Determinant conjecture. As in the sine-Gordon model, the sinh-Gordon one-point functions in the fermionic basis satisfy

$$\frac{\left\langle\boldsymbol{\beta}^\ast_{I^+}\overline{\boldsymbol{\beta}}^\ast_{\overline I^+}\overline{\boldsymbol{\gamma}}^\ast_{\overline I^-}\boldsymbol{\gamma}^\ast_{I^-}\Phi_\alpha(0)\right\rangle_R}{\left\langle\Phi_\alpha(0)\right\rangle_R}=\mathcal D\left(I^+\cup(-\overline I^+)\vert I^-\cup(-\overline I^-)\vert\alpha\right).$$

This conjecture gives a determinant expression for expectation values of arbitrary fermionic descendants in the sinh-Gordon model, extending the corresponding sine-Gordon formula. The supplied passage does not state whether the conjecture has been proved or remains open.