Chowla subspace matching conjecture

Let $K\subset L$ be a field extension, and let $A$ and $B$ be two $n$-dimensional $K$-subspaces of $L$. A $K$-subspace $B$ is called a Chowla subspace if every nonzero $b\in B$ satisfies

$$[K(b):K]\geq \dim_K B+1.$$

Chowla matching conjecture. If $B$ is a Chowla subspace, then $A$ is matched to $B$.

This is proposed as a linear counterpart to the corresponding matching theorem for Chowla subsets in abelian groups. The source notes that primitive subspaces in finite separable extensions are Chowla subspaces and gives a lower bound for the maximum possible dimension, but the asserted matching statement remains unresolved.