Non-spherical sets versus lines in Euclidean Ramsey theory

We show that for every non-spherical set $X$ in ${\mathbb{E}}^d$, there exists a natural number $m$ and a red/blue-colouring of ${\mathbb{E}}^n$ for every $n$ such that there is no red copy of X and no blue progression of length $m$ with each consecutive point at distance $1$. This verifies a conjecture of Wu and the first author.