Herremans, Huybrechs and Trefethen's lightning plus polynomial approximation conjecture

Let $0<\alpha<1$, let $N=N_1+N_2$, and define

$$r_N(x)=\sum_{j=1}^{N_1}\frac{a_j}{x-p_j}+P_{N_2}(x),$$

where $P_{N_2}$ is a polynomial of degree $N_2$ and

$$p_j=-C\exp\left(-\frac{2\pi}{\sqrt{\alpha}}\big(\sqrt{N_1}-\sqrt{j}\big)\right),\qquad 1\leq j\leq N_1.$$

Herremans, Huybrechs and Trefethen's conjecture. There exist coefficients $\{a_j\}_{j=1}^{N_1}$ and a polynomial $P_{N_2}$ with $N_2=\mathcal{O}(\sqrt{N_1})$ such that

$$|r_N(x)-x^\alpha|=\mathcal{O}\big(e^{-2\pi\sqrt{\alpha N}}\big)$$

as $N\to\infty$, uniformly for $x\in[0,1]$. This conjecture concerns the optimal convergence rate of lightning plus polynomial approximations for $x^\alpha$; the corresponding rate is proved for $\sqrt{x}$ under an additional quadrature-error assumption, while the general case $0<\alpha<1$ remains conjectural.