The cooperative two-sided secretary asymptotic rank conjecture

In the cooperative two-sided secretary game, let $N$ be the number of rounds and let $R_N(1)$ denote the expected $N$-rank of a player entering the game, under an optimal common strategy agreed on beforehand. Cooperative asymptotic rank conjecture. The asymptotics satisfy

$$\lim_{N\rightarrow\infty}\frac{R_N(1)}{\sqrt{N}}=\sqrt{27/32}\approx 0.92.$$

The paper derives a recurrence for the optimal cooperative strategy but states that the problem has not been solved completely; the conjectured constant is motivated by an approximation and computer calculations.