Geometric Erdős–Hajnal conjecture for induced restrictions

Let $\operatorname{PG}(k-1,q)$ and $\operatorname{PG}(n-1,q)$ be projective geometries, identified with their point sets. For $H\subseteq\operatorname{PG}(k-1,q)$ and $X\subseteq\operatorname{PG}(n-1,q)$, say that $X$ contains $H$ as an induced restriction if there is an injective homomorphism $\psi:\operatorname{PG}(k-1,q)\to\operatorname{PG}(n-1,q)$ such that, for every point $x$, $\psi(x)\in X$ if and only if $x\in H$.

Geometric Erdős–Hajnal conjecture. For every prime power $q$ and all $k\in\mathbb N$ and $H\subseteq\operatorname{PG}(k-1,q)$, there exist $\delta,C>0$ such that, for all $n\in\mathbb N$ and $X\subseteq\operatorname{PG}(n-1,q)$ not containing $H$ as an induced restriction, there is a subspace $W$ of $\operatorname{PG}(n-1,q)$ with $\dim(W)>Cn^\delta$ such that either $W\subseteq X$ or $W\cap X=\varnothing$.

This is the induced-restriction, two-colour specialization of the multicoloured geometric conjecture. Its general validity is open, and the source notes a stronger linear version as a further conjecture.