Let $E$ be a Banach space. Recall that the collection of all closed linear subspaces of $E$ can be turned into a metric space in a number of ways. In particular, consider the notion of a gap: if $G$ and $H$ are subspace of $E$, then $$g(G,H)=\max\{\sup\limits_{g\in \partial B_{G}} d(g, H),~\sup\limits_{h\in \partial B_{H}} d(h, G)\},$$ where by $\partial B_{G}$ and $\partial B_{H}$ we mean the intersection of the unit sphere with $G$ and $H$, respectively. Note, that $g(G,H)\le h(G,H)\le 2g(G,H)$, where $h(G,H)$ is the Hausdorff distance between $\partial B_{G}$ and $\partial B_{H}$.

Let $F\subset G$ be subspaces of $E$ and let $\varepsilon>0$. Does there exist $\delta>0$ such that every subspace $H$ with $g(G,H)<\delta$ contains a sub-subspace $J\subset H$ with $g(F,J)<\varepsilon$?