Optimal tapered lightning approximation conjecture for $x^\alpha$

Let $\alpha$ be the exponent in the approximation of $x^\alpha$ on $[0,1]$, and let $r(x)$ be a lightning + polynomial approximation with $N_1$ lightning terms, $N_2=\mathcal{O}(\sqrt{N_1})$, 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 $b(x)$ such that, for $\sigma=2\pi/\sqrt{\alpha}$, $r(x)$ satisfies

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

as $N\to\infty$, uniformly for $x\in[0,1]$. This conjecture is motivated by balancing the truncation error with the minimax approximation error; numerical experiments indicate that the predicted spacing parameter achieves the best approximation rate.