Mubayi's conjecture on d-cluster-free families of subspaces

Let $V$ be an $n$-dimensional vector space over $\mathbb{F}_q$, and let $k \ge d \ge 3$ be integers satisfying

$$n \ge \frac{dk}{d-1}.$$

For a family $\mathcal{F} \subset \genfrac{[}{]}{0pt}{}{V}{k}$, call it $d$-cluster-free if it contains no $d$ members whose sum has dimension at most $2k$ and whose intersection is $\{\mathbf{0}_V\}$. Mubayi's conjecture. If $\mathcal{F}$ is $d$-cluster-free, then

$$|\mathcal{F}| \le \genfrac{[}{]}{0pt}{}{n-1}{k-1}_q,$$

with equality only if $\mathcal{F}$ is a maximum-sized star. This is the vector-space analogue of the corresponding set-system conjecture; the asserted extremal bound and equality characterization remain open in this generality.