Kontsevich–Suhov inverse-gamma conjecture for the diagonal distribution

Let $\mu_c$ be the conjectural family of measures with values in $\operatorname{Det}^c$, let $\nu_c$ be its induced measure on welding homeomorphisms, and let $a$ be the diagonal parameter in a triangular factorization. Set $H=-\log(a)$, $\alpha=1-c$, and choose $\beta_c>0$; the source allows that $\beta_c$ may be proportional to $h^+(c)$, the larger of two conformal-anomaly values corresponding to $c<1$. Kontsevich–Suhov inverse-gamma conjecture. The distribution of $H$ should be inverse gamma with parameters $\alpha$ and $\beta_c$, equivalently, for $x>0$,

$$\nu_c(\{\sigma:\exp(-x)\le a(\sigma)\le1\})=\frac{\Gamma(\alpha,\beta_c/x)}{\Gamma(\alpha)}.$$

Its Laplace transform should be

$$\int a^\lambda\,d\nu_c(\sigma)=\frac{2(\beta_c\lambda)^{\frac{\alpha}{2}}}{\Gamma(\alpha)}K_\alpha\big(\sqrt{4\beta_c\lambda}\big),$$

where $K_\alpha$ is a modified Bessel function; this function satisfies $\lambda f”(\lambda)+c f'(\lambda)-\beta_c f(\lambda)=0$. The conjecture generalizes the proposed $c=0$ diagonal distribution, but the source gives no resolution.