The finite-subset covering conjecture for direct products of complete graphs

For a positive integer $l$, let $G_l$ have vertex set $\mathbb{N}_0^l$ and an edge between two sequences precisely when they differ in every coordinate. Let $X\subset\mathbb{N}_0^l$ be finite.

Finite-subset covering conjecture. There exist sets $X_1,\dots,X_l\subseteq X$ whose union is $X$, such that for every $i$, either $X_i$ is contained in a hyperplane of the form $\{x_i=c\}$ or the induced graph $G_l[X_i]$ is connected.

This is presented as another version of the monochromatic component covering conjecture. The supplied text does not give a resolution status for this formulation, so it remains open in the database.