Direct formula conjecture for the multitype derivative-martingale limit

Let $\mathcal{N}_t^2$ be the set of type-$2$ particles at time $t$, with positions $X_u(t)$, and let $\overline Z_\infty$ denote the previously defined limit of the relevant multitype branching-Brownian-motion martingale. Direct formula conjecture. Almost surely,

$$\lim_{t \to \infty} \sum_{u \in \mathcal{N}_t^2} (\sqrt{2}t-X_u(t))e^{\sqrt{2}X_u(t)-2t}=\overline Z_\infty.$$

The formula is proposed as a direct computation of $\overline Z_\infty$ from a submartingale. Its status is unresolved in the supplied text.