Bifurcation measure as the limiting root measure

Determine whether the bifurcation measure ν_bif constructed from the Lyapunov exponent of the Burau representation of random positive 3-braids gives the limiting distribution of roots of Alexander polynomials for closures of such braids; i.e., prove that ν_bif equals the almost-sure weak limit of ν_{w_n} as n→∞.

Background

The authors define a Lyapunov exponent λ(t) for the random products of Burau matrices evaluated at t and introduce a bifurcation measure ν_bif via χ(t)=max{λ(t),log^+|t|}. They then prove equidistribution of roots toward ν_bif on a subset U of the plane and conjecture ν_bif coincides with the full limiting measure of roots.

Establishing ν_bif as the global limiting measure would unify dynamical properties of the Burau representation with the statistical distribution of Alexander polynomial roots and fully resolve Conjecture 1 with an explicit candidate for ν∞.