The stronger q-analogue of the reverse oddtown theorem

Let $n$ be an even positive integer and let $F_q^n$ be an $n$-dimensional vector space over the finite field with $q$ elements. Write $\mathrm{sub}(F_q^n)$ for its subspaces, and let $[n-1]_q$ denote the $q$-integer. Let $\mathcal{F} \subseteq \mathrm{sub}(F_q^n)$ satisfy that $\dim(A)$ is even for every $A \in \mathcal{F}$ and $\dim(A \cap B)$ is odd for every distinct $A,B \in \mathcal{F}$. The stronger q-analogue of the reverse oddtown theorem. If $q$ is an odd prime power, then

$$|\mathcal{F}| \leq [n-1]_q.$$

The preceding reverse oddtown q-analogue gives the weaker bound $|\mathcal{F}| \leq [n]_q-1$ when $n$ is even, while the family of codimension-one subspaces inside a fixed $(n-1)$-dimensional subspace has size $[n-1]_q$. The conjecture asserts that this construction is optimal; its status is open based on the information provided.