More on r-cross t-intersecting families for vector spaces

Let $V$ be a finite dimensional vector space over a finite field. Suppose that $\mathscr{F}_1$, $\mathscr{F}_2$, $\dots$, $\mathscr{F}_r$ are $r$-cross $t$-intersecting families of $k$-subspaces of $V$. In this paper, we determine the extremal structure when $\prod_{i=1}^r|\mathscr{F}_i|$ is maximum under the condition that $\dim(\bigcap_{F\in\mathscr{F}_i}F)<t$ for each $i$.