The universal-rank-symmetry conjecture for the two-sided secretary problem

In the two-sided secretary game, suppose there are $U$ men and $U$ women, each player meets $N$ partners over $N$ rounds, and preferences satisfy universal rank symmetry: if a man is ranked $R$th in a woman's universal ranking, then she is ranked $R$th in his universal ranking. Let $R_N(1)$ denote the expected $N$-rank of a player entering the game under an optimal strategy. Universal-rank-symmetry conjecture. The asymptotics satisfy

$$\lim_{N\rightarrow\infty}R_N(1)=\text{a constant}<5.$$

The paper poses the optimal-strategy problem in this symmetric setting and conjectures that universal rank symmetry changes the outcome so that the expected $N$-rank remains bounded by a constant below $5$; the claim is presented as an open problem.