Prime-degree linear acyclic matching conjecture for field extensions

Let $K\subset L$ be a field extension, and say that it has the linear acyclic matching property if every pair of nonzero $n$-dimensional $K$-subspaces $A,B\subset L$ with $AB\cap A=\{0\}$ admits an acyclic matching from $A$ to $B$.

Prime-degree linear acyclic matching conjecture. There are infinitely many primes $p$ for which there exists a field extension $K\subset L$ with $[L:K]=p$ such that $K\subset L$ does not have the linear acyclic matching property.

The linear matching property has a known classification, whereas the corresponding acyclic property is not classified. The purely transcendental case is known, but the behavior of prime-degree extensions and finite extensions without proper intermediate fields remains open.