The first-order exit-distribution conjecture for symmetric smart kinetic walks

Let $D$ be a simply connected domain containing the origin, and let $\delta$ be the lattice spacing of the square lattice. Let $\mu_\delta(0,|dz|;D)$ be the discrete exit distribution of a smart kinetic walk with symmetric transition probability, defined using orthogonal projection of the first point outside $D$ onto $\partial D$. Let $\mu(0,|dz|;D)$ be harmonic measure, and let $|dz|$ denote Lebesgue measure on the boundary with respect to arc length. First-order symmetric-SKW conjecture. There is a function $\rho_D(z)$ on $\partial D$ depending only on $D$ and, for each such SKW, a constant $c$ depending only on the transition probability, such that

$$\lim_{\delta\to 0}\frac{\mu_\delta(0,|dz|;D)-\mu(0,|dz|;D)}{\delta}=c\rho_D(z)|dz|.$$

The conjecture refines the reported numerical convergence of the symmetric SKW exit distribution to harmonic measure by predicting its first-order discrepancy. The simulations support the asserted domain-independence of $\rho_D$ and transition-probability dependence of $c$, but the source provides no proof.