$q$-analogue of the Erdős–Chvátal simplex conjecture

Let $\mathbb{F}_q^n$ be the $n$-dimensional vector space over the finite field with $q$ elements, and let a $d$-simplex of $k$-dimensional subspaces mean the corresponding configuration with empty total intersection and nonempty intersection for every $d$-member subcollection. $q$-analogue Erdős–Chvátal simplex conjecture. If $k\geq d+1\geq3$, $n\geq\frac{k(d+1)}{d}$, and $\mathcal{V}$ is a family of $k$-dimensional subspaces of $\mathbb{F}_q^n$ with no $d$-simplex, then

$$|\mathcal{V}|\leq q^{n-k}\genfrac{[}{]}{0pt}{}{n-1}{k-1}=\genfrac{[}{]}{0pt}{}{n}{k}-\genfrac{[}{]}{0pt}{}{n-1}{k}.$$

The paper explicitly presents this as an open problem and as a $q$-analogue of the Erdős–Chvátal simplex conjecture.