Conjectured asymptotic for random Haar unitary pairs: lim c(U^{(N)}, u0) = √2

Establish that for U^{(N)} a pair of independent Haar-distributed N×N unitaries and u_0 the universal 2-tuple of commuting unitaries, the dilation constant c(U^{(N)}, u_0) converges almost surely to √2 as N → ∞.

Background

The authors present extensive numerical evidence that c(U^{(N)}, u_0) approaches √2 as N increases. Under their limit theorems, such convergence would imply c(u_f, u_0) = √2 and, combined with c(u_u, u_f) = 2/√3, would yield the conjectured bound C_2 ≤ 2√(2/3).

A rigorous proof would connect random matrix asymptotics to operator system dilation constants and solidify the empirical picture of commuting dilations for free unitaries.