The reproducing-kernel asymptotics conjecture for weighted residual polynomials on Jordan arcs

Let $K$ be a $C^{1+}$ Jordan arc, let $z_0 \in \overline{\mathbb{C}}\backslash K$, and let $\rho:K\to[0,\infty)$ be upper-semicontinuous. Writing $t_n(\rho,z_0)$ for the weighted residual-polynomial quantity, $C(K,z_0)$ for the associated condenser-capacity factor, $S(\rho,z_0)$ for the weight-dependent factor, and $K_{\omega_{z_0}}$ for the reproducing kernel associated with the harmonic measure $\omega_{z_0}$, the reproducing-kernel asymptotics conjecture. The limit

$$\lim_{n\to\infty}\frac{t_n(\rho,z_0)}{C(K,z_0)^n}=\frac{S(\rho,z_0)}{K_{\omega_{z_0}}(z_0,z_0)}$$

should hold. The formula is known for circular arcs, while its validity for general sufficiently smooth Jordan arcs remains open.