Straight-line motion conjecture for heat-flow zeros

Let $T_t:\mathbb{C}\to\mathbb{C}$ be the deterministic transport maps associated with the hydrodynamic limit, let $t_{\mathrm{sing}}$ be the first singular time, and let $z_j(t)$ denote the zeros of the heat-evolved polynomial. Straight-line motion conjecture. In the hydrodynamic limit, for $0<t<t_{\mathrm{sing}}$, the motion of each individual zero is described by $T_t$; when $n$ is large, $z_j(t)$ stays with high probability close to the curve $T_t(z_j(0))$, and the roots approximately move with constant velocity along straight lines. This conjecture refines the transport-map picture before singularities form, where the Stieltjes-transform dynamics support approximately constant velocities.