$s$-almost $t$-intersecting families for vector spaces

Let $\mathcal{F}$ be a family of $k$-dimensional subspaces of an $n$-dimensional vector space. Write ${\mathcal{D}}_{\mathcal{F}}(H;t)=\{F\in \mathcal{F}\text{\hspace{0.17em}⁣}:\dim (F\cap H)\le t\}$ for a subspace $H$. The family $\mathcal{F}$ is called $s$-almost $t$-intersecting if $\mathrm{\mid }{\mathcal{D}}_{\mathcal{F}}(F;t)\mathrm{\mid }\le s$ for each $F\in \mathcal{F}$. In this note, we prove that $s$-almost $t$-intersecting families with maximum size are $t$-intersecting.