Mad families of Gowers' infinite block sequences

Call a subset of ${\mathbf{F}\mathbf{I}\mathbf{N}}_k$ small if it does not contain a copy of $⟨A⟩$ for some infinite block sequence $A\in {\mathbf{F}\mathbf{I}\mathbf{N}}_k^{[\mathrm{\infty }]}$. Gowers’ ${\mathbf{F}\mathbf{I}\mathbf{N}}_k$ theorem asserts that the set of small subsets of ${\mathbf{F}\mathbf{I}\mathbf{N}}_k$ forms an ideal, so it is sensible to consider almost disjoint families of ${\mathbf{F}\mathbf{I}\mathbf{N}}_k$ with respect to the ideal of small subsets of ${\mathbf{F}\mathbf{I}\mathbf{N}}_k$. We shall show that ${\mathfrak{a}}_{{\mathbf{F}\mathbf{I}\mathbf{N}}_k}$, the smallest possible cardinality of an infinite mad family of ${\mathbf{F}\mathbf{I}\mathbf{N}}_k$, is uncountable.