Gopalan–Klivans–Zuckerman conjecture on Reed–Muller list-decoding radius

Let $d$ be fixed and let $\mathbb{K}$ be a finite field. For $\mathbb{K}$, let $textnormal{LDR}_{\mathbb{K}}(n,d)$ denote the list-decoding radius of the Reed–Muller code, and let $\delta_{\mathbb{K}}(d)$ denote its normalized minimum distance, defined by writing $d=a(q-1)+b$ with $0\leqslant b<q-1$ and setting

$$\delta_{\mathbb{K}}(d)=\frac{1}{q^a}\left(1-\frac{b}{q}\right),$$

where $q=|\mathbb{K}|$.

Gopalan–Klivans–Zuckerman conjecture. For fixed $d$ and finite field $\mathbb{K}$,

$$\textnormal{LDR}_{\mathbb{K}}(n,d)=\delta_{\mathbb{K}}(d).$$

The right-hand side is the natural upper bound coming from the normalized minimum distance of Reed–Muller codes. Gopalan, Klivans, and Zuckerman proved the equality over $\mathbb{F}_2$; the conjecture asserts that the matching lower bound holds over every finite field.