Convergence of the spectral radius to one

Prove that the spectral radius ρ(Y_n) of the self-normalized random matrix converges to one, by complementing the established asymptotic upper bound ρ(Y_n)≤1+ε in probability with a matching lower bound.

Background

The paper proves that for every ε>0, the probability that the spectral radius of Y_n exceeds 1+ε tends to zero. Thus, the spectral radius is asymptotically bounded above by one in probability.

A matching lower bound would imply convergence of ρ(Y_n) to 1. The authors explicitly formulate this as a conjecture, leaving unresolved whether the spectral radius approaches the unit-circle scale from below.