Some context.
Let $A$ and $B$ be two subspaces of dimension $k$ of $\mathbb R^{2k}$. Let's define $\psi(A,B)$, the sinus of the angle between $A$ and $B$ as
$$\psi(A,B)=\min_{\substack{a\in A\setminus\{0\}\\ b\in B\setminus\{0\}}} \vert\sin\widehat{(a,b)}\vert.$$
We can notice that the angle between a vector $b$ and a subspace $A$ is
$$\psi(b,A)=\vert\sin\widehat{(b,p_A(b))}\vert$$
where $p_A$ is the orthogonal projection onto $A$.
What we know.
We know that there exists $b_1,\ldots,b_k\in B$ linearly independents such that
for all $i\ne j$, $\widehat{(b_i,b_j)}\geqslant \pi/4$ ;
there exists a constant $c_1>0$ such that
$$\forall i\in\{1,\ldots,k\},\quad \vert\sin\widehat{(b_i,p_A(b_i))}\vert\geqslant c_1$$
where $p_A$ is the orthogonal projection onto $A$.
What we want to prove.
Can we prove that: if $c_1$ is sufficiently small, there exists a constant $c_2$ (depending on $c_1$) such that
$$\psi(A,B)\geqslant c_2 \quad?$$
Final remarks.
I haven't made any progress yet, so any hints, references or ideas would be much appreciated.