Matching conjecture for the stochastic Bessel limiting point process

For $a\geq 0$, let $\mathcal{M}_a=\sum_{i\geq 1}\delta_{\mu_a^{\infty}(i)}$ be the limiting point process, and let $(g_j)_{j\geq 1}$ be independent exponential random variables with parameters $j(j+a)/2$. Define

$$\hat{\mu}_a^{\infty}(k):=\sum_{j=k}^{\infty}g_j,\qquad k\geq 1.$$

Matching conjecture. For all $a\geq 0$, the joint distributions of $(\hat{\mu}_a^{\infty}(i),\;i\geq 1)$ and $(\mu_a^{\infty}(i),\;i\geq 1)$ are equal. As a corollary, using the explicit expression for the density of $\hat{\mu}_a^{\infty}(1)$,

$$\sum_{j=1}^{\infty}(-1)^{j-1}\frac{j(2j+a)}{2}\binom{j+a}{j}e^{-\frac{j(j+a)}{2}x},$$

this matching yields

$$\mathbb{P}\big(\exists t\geq 0,\;R^{-a}(t)\geq\mu+bt\big)=\sum_{j=1}^{\infty}(-1)^{j-1}\left[\binom{j+a/b}{j}+\binom{j+a/b-1}{j-1}\right]e^{-2\mu j(jb+a)},$$

where $R^{-a}$ is a reflected Brownian motion with drift $-a$.

The conjecture extends the proved equality of the largest points at $a=0$ to the entire point process and all $a\geq 0$. If true, it would provide an unexpected diffusion representation of the exponential-gap construction and an exact formula for the hitting probability of a reflected Brownian motion with negative drift.