Tropical cyclic rotation agrees with tableau promotion

Let $\operatorname{SSYT}(k,[n])$ be the semistandard Young tableaux of rectangular shape, let $\rho$ be the tropical cyclic-rotation map on tableaux, and let $\operatorname{pr}$ be promotion, defined by the Bender–Knuth involutions.

Promotion conjecture. For any tableau $T\in\operatorname{SSYT}(k,[n])$ without frozen factors,

$$\rho(T)=\operatorname{pr}(T).$$

This conjecture identifies tropical cyclic rotation with the classical promotion operation on tableaux, after frozen factors are removed. The supplied text gives no resolution status.