Linear analogue of the product-set dimension inequality

Let $K\subset L$ be a field extension, and let $A,B\subset L$ be finite-dimensional $K$-subspaces of positive dimension. Suppose that for every $a,b\in L$ and every nontrivial proper finite intermediate subfield $K\subset M\subsetneq L$, one has

$$\dim_{K}(aM\cap A)+\dim_{K}(bM\cap B)\leq [M:K]+1.$$

Linear product-set conjecture. Then every pair of $K$-subspaces $S\subset A$ and $T\subset B$ satisfies

$$\dim_{K}\langle ST\rangle\geq \dim_{K}S+\dim_{K}T-1.$$

This is proposed as a linear analogue of the set-theoretic inequality underlying the matching result in Theorem 1(7), extending the role of the corresponding additive-combinatorial proposition to field extensions.