Conjecture on the conditioned critical mass distribution

Let $X_t$ be the mass variable of the continuous-time Derrida–Retaux model started from $\mu_0$, and let $\mathcal{L}(X_t\mid X_t>0)$ denote its conditional law given positivity. Conditioned-law conjecture. If

$$F_\infty(\mu_0)=0,$$

then $\mathcal{L}(X_t\mid X_t>0)$ converges to an exponential distribution with parameter at least $1$. This is motivated by the differential equation for a possible limiting density, but the source gives no resolution and presents the statement as a conjecture.