Abrikosov's triangular-lattice conjecture for beta-exponent planar zero packing

For a positive real $\beta$, let $\rho_\beta(\mathbb{C})$ denote the $\beta$-exponent minimal discrepancy density for planar zero packing, defined by

$$\rho_\beta(\mathbb{C})=\liminf_{R\to+\infty}\inf_f\frac{1}{R^2}\int_{\mathbb{D}(0,R)}\big(|f(z)|^\beta\mathrm{e}^{-|z|^2}-1\big)^2\,\mathrm{d} A(z),$$

where the infimum is over all polynomials $f$. Abrikosov's conjecture. The equilateral triangular lattice is optimal for $\beta$-exponent planar zero packing for each positive $\beta$. The equilateral triangular lattice is known to be optimal among lattices when $\beta=2$, but the asserted optimality for every positive $\beta$ remains open.