Asymptotic prevalence of zero diagonal entries among permutation matrices

Ascertain whether lim_{d→∞} |C_d|/d! = 1, where C_d ⊂ Cont_{1^d,1^d} is the set of d×d permutation matrices a for which the diagonal entry RSK_{1^d,1^d}(a,a) equals 0.

Background

For permutation weights, Proposition 4.1 expresses diagonal entries as products of determinants of partial permutation matrices. Empirical counts of C_d, the set where the diagonal entry vanishes, grow rapidly with d. The conjecture predicts that almost all permutations yield zero diagonal entries asymptotically.