Conjectural diagonal distribution for Werner's measure

Let $\nu_0$ be the probability measure induced by Werner's loop measure on the triangular factorization parameter $a$ of a welding homeomorphism, and let $\beta_0<\frac{5\pi^2}{4}$. Diagonal-distribution conjecture. For every $x>0$, one should have

$$\nu_0(\{\sigma:\exp(-x)\le a(\sigma)\le1\})=\exp\left(-\frac{\beta_0}{x}\right).$$

This specifies the proposed diagonal distribution and its Laplace transform; the source gives no resolution of the conjecture.