Lower bound for the minimum distance of complete intersections

Let $X=CI(d_1,\ldots,d_n)\subseteq\mathbb{P}^{n}$ be a complete intersection, with

$$2\leq d_1\leq\cdots\leq d_n.$$

Complete-intersection distance conjecture. Then

$$d(X)\geq (d_1-1)d_2d_3\cdots d_n.$$

The conjecture asserts that the hypothesis in the preceding corollary—that one of the associated complete-intersection curves has no component contained in a hyperplane—can be dropped. It gives a uniform lower bound for the minimum distance invariant of complete intersections, but the source provides no resolution.