Extremal bound for suboptimal $(2d+1)$-union antichains

Let $V$ be an $n$-dimensional vector space over the finite field with $q$ elements, and let $\mathcal{L}(V)$ denote its lattice of subspaces. A family $\mathcal{F}\subseteq\mathcal{L}(V)$ is a $(2d+1)$-union antichain if it is an antichain whose union of any $2d+1$ members is not a member of the family. Assume that $\mathcal{F}$ is not contained in any optimal $(2d+1)$-union antichain and that $2d+1<n$. Let $\left[n\atop d\right]$ denote the Gaussian binomial coefficient and let $\mathcal{B}[n,2d+1]$ be the family defined in the source. Extremal bound conjecture.

$$|\mathcal{F}|\leq \left[n\atop d\right]-q\left[d\atop 1\right].$$

Moreover, equality holds if and only if $\mathcal{F}=\mathcal{B}[n,2d+1]$. This conjecture seeks the sharp size bound and equality characterization for suboptimal $(2d+1)$-union antichains; the notation $\mathcal{B}[n,2d+1]$ is not defined in the supplied context, and its structure and the conjecture's resolution require verification.