The absorption-time and Brownian-bridge maximum distribution conjecture

For each $N\geq 1$, let $T_{2N}$ and $T_{2N+1}$ be the absorption times in the discrete Whittaker process. Let $M_N$ be the maximum height of $N$ non-intersecting reflected Brownian bridges starting and ending at zero, and let $H_N$ be the maximum height of $N$ non-intersecting Brownian excursions. Absorption-time distribution conjecture.

$$T_{2N}\stackrel{d}{=}2(M_N)^2,\qquad T_{2N+1}\stackrel{d}{=}2(H_N)^2.$$

The conjecture would connect absorption times of the discrete Whittaker process with maxima of non-intersecting Brownian paths; it is consistent with the explicitly verified cases $N=1$ and $N=2$. The cited asymptotic results for $M_N$ and $H_N$ would imply Tracy–Widom GOE fluctuations for the absorption times if the conjecture holds.