Limiting first-descent law for rectangular tableaux with k rows

Determine the limiting distribution of the first descent and the limiting parity constant for uniformly random standard Young tableaux of rectangular shape (n^k) as n tends to infinity, for general fixed k.

Background

The paper’s decomposition for shape (n,n,n) extends to rectangular shapes (n^k), using the identity f^{(n^k)/(r,1)}=f^{(n^{k-2},n-1,n-r)}. For three rows, the paper proves that the first-descent distribution converges and that the probability of an odd T(2,1) tends to 27/64.

The cases k=1 and k=2 are described as empty or easy, respectively, while the unrestricted-shape comparison has limiting parity constant 1/e. The general interpolation across the number of rows remains unresolved.