Theoretical explanation for the growth of the dominant polynomial root

Establish whether there is a theoretical reason why the largest real root r^{(ell)} of H_ell(log(z)/pi) increases by approximately a factor of three as ell increases.

Background

The proof of eventual alternation uses the largest real root r^{(ell)} of the transformed polynomial H_ell(log(z)/pi). Numerical evidence indicates that this root grows by roughly a factor of three with each increase in ell, paralleling the observed behavior of the threshold \Gr(ell).

The paper gives a natural but comparatively large upper bound for r^{(ell)}, whose asymptotic growth rate is about 3.606. The authors explicitly ask for a theoretical explanation of the apparently smaller tripling rate.