Tohăneanu–Van Tuyl conjecture on the minimum distance of complete intersections

Let $\mathbb{X}$ be a finite set of points in $\mathbb{P}^{s-1}$, and let $I(\mathbb{X})$ be its vanishing ideal. Suppose that $I(\mathbb{X})$ is a complete intersection generated by $f_1,\ldots,f_{s-1}$, where $e_i=\deg(f_i)$ for $i=1,\ldots,s-1$ and $2\leq e_i\leq e_{i+1}$ for all $i$. The minimum distance $\delta_\mathbb{X}(1)$ of the corresponding projective Reed–Muller-type code is the minimum distance in degree one. Tohăneanu–Van Tuyl conjecture. One has

$$\delta_\mathbb{X}(1)\geq (e_1-1)e_2\cdots e_{s-1}.$$

This conjecture gives a lower bound for the minimum distance of evaluation codes on complete intersections and is related to whether the uniform bound for the number of zeros remains valid when the vanishing ideal itself, rather than its initial ideal, is assumed to be a complete intersection. The source does not state a resolution, so the conjecture is recorded as open.