Step-count conjecture for the switching algorithm

Let $(\boxtimes,T)\in \operatorname{LR}(\alpha,\scalebox{.5}{\yng(1)},\beta,\gamma)$, where $\operatorname{LR}(\alpha,\scalebox{.5}{\yng(1)},\beta,\gamma)$ is the set of type B Littlewood–Richardson objects used by the switching algorithm. Let $s$ be the index of the first Phase 2 step, and let $\operatorname{esh}(\boxtimes,T)$ be the associated elementary switching operation. Step-count conjecture. The number of steps (switches) in the switching algorithm for computing $\operatorname{esh}(\boxtimes,T)$ is equal to

$$2s+\beta_s-1.$$

In particular, there are exactly as many steps as in the promotion step of the rectification algorithm described in the paper. This is presented as a surprising combinatorial conjecture and is not proved in the source.