Conjecture that partitionable convergence is equivalent to left-convergence

Let $(\mathscr{P}_n)$ be a sequence of branching partitions and let $\widetilde{\mathscr{P}}$ be a branching partition. Write $\mathscr{P}_n \dashrightarrow \widetilde{\mathscr{P}}$ for partitionable convergence and $\mathscr{P}_n \to \widetilde{\mathscr{P}}$ for left-convergence. Partitionable convergence conjecture.

$$\mathscr{P}_n \to \widetilde{\mathscr{P}} \quad\text{if and only if}\quad \mathscr{P}_n \dashrightarrow \widetilde{\mathscr{P}}.$$

This conjecture proposes a converse to the lemma establishing that left-convergence implies partitionable convergence. It asserts that the two notions of convergence for branching partitions are equivalent.