Suppose $V$ is a symplectic, unitary, or orthogonal space over a finite field $\mathbb{F}_q$ of characteristic $p$. Suppose $H\subseteq U\subseteq W$ is a chain of subspaces of $V$, where $\dim U=\dim H+1$, $\dim W=\dim U+1$, and $U$ is non-degenerate.
Question: Is there always a subspace $U'\subseteq W$ isometric to $U$ such that $U'\cap U=H$, assuming $\dim U\geq c$ for some large enough constant $c$?
I need this statement for a problem related to finite classical groups. I did some calculations and feel that the statement seems to be true, but the details are getting tedious. For symplectic and unitary spaces I think I can quickly prove it (by discussing the cases $W$ is non-degenerate / has a nonzero radical). For orthogonal spaces, there are more cases and I don't have clear ideas. The case $p=2$ seems particularly subtle as one can have isotropic but nonsingular $v\in W$ orthogonal to every vector in $W$.
Can someone help me verify the statement, or suggest a good way of proving it, or tell me if this statement is already known in some papers? Many thanks.