Optimal tapered lightning approximation conjecture for $z^\alpha$ on a V-shaped domain

Let $\alpha$ be the exponent, let $\beta\in[0,2)$, and consider the V-shaped domain $[0,1]e^{\pm i\beta\pi/2}$. Let $r(z)$ be a lightning + polynomial approximation with $N_1$ lightning terms, a polynomial part of degree $N_2=\mathcal{O}(\sqrt{N_1})$, and tapered lightning poles with spacing parameter $\sigma$. Optimal tapered lightning approximation conjecture. There exist coefficients $\{a_j\}_{j=1}^{N_1}$ and a polynomial $b(z)$ such that, with

$$\sigma=\frac{\sqrt{2(2-\beta)}\pi}{\sqrt{\alpha}},$$

one has

$$\lvert r(z)-z^\alpha\rvert=\mathcal{O}\left(e^{-\pi\sqrt{2(2-\beta)\alpha N}}\right)$$

as $N\to\infty$, uniformly for $z\in[0,1]e^{\pm i\beta\pi/2}$ and $\beta\in[0,2)$. Numerical experiments indicate that the corresponding spacing parameter is optimal, generalizing the observed result for $\sqrt z$; the claim remains unproved.