Chui's equispaced-poles conjecture in the Bergman space

Let $\mathbb D$ be the unit disk and let $L^1(\mathbb D)$ denote the space of integrable functions on $\mathbb D$ with respect to normalized planar Lebesgue measure. For a positive integer $N$ and points $a_0,\ldots,a_{N-1}$ on the unit circle, consider the simplest fraction $\sum_{0\le k<N}(z-a_k)^{-1}$. Chui's conjecture. For every positive integer $N$ and every family of points $\{a_k\}_{0\le k<N}$ on the unit circle,

$$\left\|\sum_{0\le k<N}\frac{1}{z-a_k}\right\|_{L^1(\mathbb D)}\ge \left\|\sum_{0\le k<N}\frac{1}{z-e^{2\pi i k/N}}\right\|_{L^1(\mathbb D)}.$$

Equivalently, among simplest fractions with $N$ poles on the unit circle, the one with equispaced poles has minimal $L^1(\mathbb D)$ norm. The paper's abstract states that this conjecture is solved for a wide class of weighted Hilbert Bergman spaces, while the displayed $L^1$ formulation is presented as Chui's original conjecture; the supplied text does not state that this exact formulation has been resolved.