The asymptotic degree of the bubble-sort map on partitions

Let $\operatorname{Part}(n)$ denote the set of partitions of $n$, and let $\mathscr B:\operatorname{Part}(n)\to\operatorname{Part}(n)$ be the bubble-sort map. Write $\deg(\mathscr B:\operatorname{Part}(n)\to\operatorname{Part}(n))$ for its maximum fiber size.

Bubble-sort degree conjecture.

$$\lim\limits_{n\to\infty}\deg(\mathscr B:\operatorname{Part}(n)\to\operatorname{Part}(n))=3.$$

Computations for random partitions suggest that the degrees are close to $3$, but the paper states that no method is currently known to improve the preceding estimates.