Martinec's annulus bootstrap conjecture for boundary Liouville CFT

Consider an annulus represented as a cylinder of length $\pi\tau$ and radius $1$. Set $q=e^{-2\pi\tau}$, and let $\mathcal{Z}_{\mathrm{Annulus}}$ denote the no-insertion-point partition function of Liouville conformal field theory on this annulus, defined probabilistically in analogy with the one-point function. Let $Q$ be the Liouville background charge, let $U_{\mathrm{FZZ}}$ denote the FZZ boundary wave function, and let $\mu$ and $\mu_B$ be the bulk and boundary cosmological constants. Then Martinec's annulus bootstrap conjecture. the partition function should satisfy

$$\mathcal{Z}_{\mathrm{Annulus}}=\int_{\mathcal{C}}U_{\mathrm{FZZ}}(Q+iP)U_{\mathrm{FZZ}}(Q-iP)q^{P^2-\frac{1}{24}}\prod_{n\geq 1}(1-q^n)\,dP.$$

The two boundary components may have different values of $\mu_B$, in which case the two FZZ wave functions are evaluated using the respective boundary cosmological constants; the bulk parameter $\mu$ is the same in the partition function and both wave functions. The contour $\mathcal{C}$ is a suitable deformation of $\mathbb{R}$ avoiding the pole at $P=0$. This is a proposed conformal-bootstrap formula for the annulus partition function in Liouville CFT; its validity is presented as a conjecture, with no resolution supplied here.