Limiting root-measure conjecture for random positive 3-braids

Let $\mathrm{Br}_3$ be the 3-strand braid group, let $\Omega=\{\sigma_1,\sigma_2\}^{\mathbb{N}}$ with the product measure assigning probability $1/2$ to each generator, and set $w_n=g_1\cdots g_n$. For a braid $w$, let $Z_w$ be the multiset of zeros of the Alexander polynomial of its closure and let

$$\nu_w=\frac{1}{\# Z_w}\sum_{z\in Z_w}\delta_z.$$

Limiting root-measure conjecture. There is a compactly supported measure $\nu_\infty$ on $\mathbb{C}$ such that, for almost every sequence $(g_i)_{i\in\mathbb{N}}\in\Omega$, $\nu_{w_n}$ converges weakly to $\nu_\infty$. Its support is contained in the union of the unit circle, a continuous curve from $\zeta_3$ to $\overline{\zeta}_3$, and the image of that curve under $z\mapsto1/z$; the curve crosses the real axis at about $-0.78$. Writing

$$\mathcal A_L=\{e^{i\theta}:|\theta-\pi|<\pi/3\},\qquad \mathcal A_R=\{e^{i\theta}:|\theta|<2\pi/3\},$$

exactly $2/3$ of the mass lies on $\mathcal A_R$, where the measure is a multiple of Lebesgue measure; its mass on $\mathcal A_L$ is $(7-3\sqrt5)/12\approx0.024316$, so the total mass on the unit circle is $(5-\sqrt5)/4\approx0.690983$. The measures along the curves joining $\zeta_3$ to $\overline{\zeta}_3$ are absolutely continuous with respect to Lebesgue measure. These predictions refine the experimentally observed concentration and equidistribution of roots; the convergence and the detailed support and density assertions remain conjectural.