Conjecture on extremal sunflower-free families of vector spaces

Let $s\leq k$ and let $q$ be a prime power. An $s$-sunflower-free family of $k$-spaces over the field with $q$ elements is a family containing no $s$ members whose pairwise intersections are all equal. Let ${\mathcal B}(s,k)$ denote the construction defined earlier in the paper.

Extremal vector-space sunflower conjecture. There exists a constant $C=C(q,s)$ such that every $s$-sunflower-free family ${\mathcal F}$ of $k$-spaces over the field with $q$ elements satisfies

$$|{\mathcal F}|\leq |{\mathcal B}(s,k)|\cdot C^k.$$

The conjecture asserts that the construction ${\mathcal B}(s,k)$ is optimal up to an exponential factor in $k$. The source motivates it by noting that no better construction is known for the relevant parameters, while the optimality of the chosen nested lifted MRD-code parameters is not clear in general.