Mixture classification conjecture for stationary gap distributions

Let $(g_n)_{n\geq 1}$ be drift parameters satisfying the conditions

$$\overline{g}_{k}>\overline{g}_{N_j}\quad\text{for }k=1,\ldots,N_j-1,\quad j\geq 1,$$

for some $N_1<N_2<\cdots\to\infty$, together with the condition. For each $a\in\mathbb{R}_+$, let $\pi_a$ denote the corresponding stationary gap distribution. A probability measure $\rho$ on $\mathbb{R}_+$ defines the mixture

$$\pi_{\rho}(\mathord{\cdot})=\int_{\mathbb{R}_+}\pi_a(\mathord{\cdot})\,\rho(\mathrm{d}a).$$

Mixture classification conjecture. Under these conditions, every stationary gap distribution of an infinite system of competing Brownian particles is equal to $\pi_\rho$ for some probability measure $\rho$ on $\mathbb{R}_+$. The conjecture would classify all stationary gap distributions as mixtures of the family $(\pi_a)_{a\in\mathbb{R}_+}$.