The non-spherical obstruction conjecture for Euclidean Ramsey sets

Let $X\subset\mathbb{E}^d$ be a set, and call it non-spherical if it is not contained in the surface of a sphere of any dimension. Write $\ell_m$ for the relevant $m$-point line configuration, and write $\mathbb{E}^n\nrightarrow(X,\ell_m)$ when there is a two-colouring of $\mathbb{E}^n$ containing neither a monochromatic copy of $X$ in the first colour nor a monochromatic copy of $\ell_m$ in the second colour. Non-spherical obstruction conjecture. For every non-spherical set $X$, there exists a natural number $m$ such that

$$\mathbb{E}^n\nrightarrow(X,\ell_m)$$

for all $n$. This would establish the difficult converse direction of the proposed Ramsey characterisation for the important class of non-spherical sets; the source presents it as a first step toward the preceding conjecture.