The vector-space Szemerédi conjecture

Let $q$ be a prime power, let $\mathbb{F}_{q}^{n}$ be the $n$-dimensional vector space over $\mathbb{F}_{q}$, and call a family of its subspaces of density at least $\delta$ when its size is at least a $\delta$ proportion of the subspaces in the lattice $\mathcal{L}_{n}(q)$. Say that such a family contains an arithmetic progression of length $k$ when it contains subspaces whose dimensions form an arithmetic progression of length $k$. Vector-space Szemerédi conjecture. For every positive integer $k$ and every $\delta>0$, there exists $n$ such that every family $\mathcal{V}$ of subspaces of $\mathbb{F}_{q}^{n}$ with density at least $\delta$ contains an arithmetic progression of length $k$.

This is proposed as a vector-space analogue of Szemerédi's theorem, with progressions defined through the dimensions of subspaces. The source does not provide a resolution.