The mixture representation conjecture for stationary gap distributions

Let the diffusion coefficients satisfy $\sigma_n=1$ for every $n\in\mathbb{Z}$, and assume the conditions of Theorem 2.1 hold. For $(a,b)\in\mathbb{R}^2$, define

$$\lambda_n=2\Phi_{n+1}(g)+a+bn,\qquad n\in\mathbb{Z},$$

and let

$$\Sigma=\{(a,b)\in\mathbb{R}^2\mid \lambda_n>0\text{ for every }n\in\mathbb{Z}\}.$$

For $(a,b)\in\Sigma$, write $\pi_{a,b}=\bigotimes_{n\in\mathbb{Z}}\operatorname{Exp}(\lambda_n)$ for the corresponding stationary gap distribution, and let $\rho$ be a probability measure on $\Sigma$. Mixture representation conjecture. Every stationary gap distribution of the two-sided infinite system can be represented as

$$\int_{\Sigma}\pi_{a,b}\,\mathrm{d}\rho(a,b)$$

for some probability measure $\rho$ on $\Sigma$. The explicit product-form measures are stationary, and the conjecture asserts that their mixtures exhaust all stationary gap distributions; the source gives no resolution of this converse statement.