Involutions conjecture for statistics on three-row standard Young tableaux

Let $\operatorname{SYT}(n^3)$ denote the set of standard Young tableaux of rectangular shape $(n,n,n)$, and let $b_{r,s}$ denote the tableau statistics used in the source.

Involutions conjecture. There exist involutions $\alpha,\delta:\operatorname{SYT}(n^3)\to\operatorname{SYT}(n^3)$ with the following properties:

• $\alpha$ swaps $b_{1,1}\leftrightarrow b_{2,1}$, $b_{1,2}\leftrightarrow b_{2,3}$, $b_{1,3}\leftrightarrow b_{2,2}$ and $b_{3,2}\leftrightarrow b_{3,3}$, and it preserves $b_{3,1}$;
• $\delta$ swaps $b_{2,1}\leftrightarrow b_{2,3}-1$, and it preserves $b_{1,2}$, $b_{2,2}$ and $b_{3,2}$.

This conjecture is motivated by computational evidence for stronger symmetry in the joint distribution of the statistics $b_{r,s}$ on $\operatorname{SYT}(n^3)$. The supplied text does not report a proof or disproof, so the conjecture remains open.