The minimal red progression length in Euclidean Ramsey theory

For integers $n,k\geq 1$, let $\mathbb{E}^n$ denote $n$-dimensional Euclidean space, and let $\ell_s$ denote a set of $s$ equally spaced collinear points. Write $\mathbb{E}^n\rightarrow(\ell_s,\ell_k)$ if every red-blue coloring of $\mathbb{E}^n$ contains either a red copy of $\ell_s$ or a blue copy of $\ell_k$. The minimal-length conjecture. There is an integer $k$ such that for every integer $n$,

$$\mathbb{E}^n\not\rightarrow(\ell_3,\ell_k).$$

This conjecture asks whether three is the smallest first progression length for which some fixed blue progression length avoids the corresponding Euclidean Ramsey property in every dimension. The preceding results establish related positive and negative statements for other progression lengths, but the conjectured case $s=3$ remains open.