Steiner Tree complexity dichotomy for 1-colorful minor-closed classes

Let $\mathcal{C}$ be a $1$-colorful minor-closed class, meaning a class of $1$-colorful graphs closed under taking colorful minors. A $1$-colorful rainbow $(k\times k)$-grid is the specified $1$-colorful rainbow grid of side length $k$ in the source. Steiner Tree complexity dichotomy conjecture. Unless $\mathsf{P}\neq\mathsf{NP}$, \textsc{Steiner Tree} can be solved in polynomial time in $\mathcal{C}$ if and only if there is some $k$ such that $\mathcal{C}$ excludes the $1$-colorful rainbow $(k\times k)$-grid as a colorful minor.

This conjecture proposes a sharp polynomial-time tractability dichotomy for \textsc{Steiner Tree} on $1$-colorful minor-closed classes, refining the bidimensionality-frontier discussion. Its resolution is not given in the supplied text.