Bijective proof of major-index palindromicity

Construct, for every partition \(\lambda\) of \(N\), an explicit bijection \(\Phi:\SYT(\lambda)\to\SYT(\lambda)\) satisfying \(\maj(T)+\maj(\Phi(T))=\binom{N}{2}+b(\lambda)-b(\lambda')\) for every standard Young tableau \(T\) of shape \(\lambda\).

Background

The paper considers the major index $\maj(T)=\sum_{i\in\Des(T)}i$ on standard Young tableaux. Stanley’s $q$-analogue of the hook-length formula gives the generating polynomial for this statistic and implies that the polynomial is palindromic, but the paper notes that a bijective proof of this palindromicity is not known.

The requested bijection should pair tableaux of the same shape so that their major indices sum to the stated constant $\binom{N}{2}+b(\lambda)-b(\lambda')$. The paper observes that evacuation proves symmetry on subsets with a fixed number of descents, but does not solve the problem globally for all tableaux of a given shape.