The cardinal-invariant lower-bound conjecture for mad families of subspaces

Let $F$ be a countable field, let $\mathrm{non}(\mathcal{M})$ denote the uniformity of the meager ideal, and let $\mathfrak{a}_{\mathrm{vec},F}$ be the least cardinality of a mad family of subspaces over $F$. Lower-bound conjecture.

$$\mathrm{non}(\mathcal{M})\leq\mathfrak{a}_{\mathrm{vec},F}.$$

Consequently, it is consistent that $\mathfrak{a}<\mathfrak{a}_{\mathrm{vec},F}$, where $\mathfrak{a}$ is the least cardinality of a mad family on $\omega$. The conjecture would remove the block-subspace assumption from the corresponding result and would yield a consistency separation between these cardinal characteristics.