The idempotent bound for affine subspaces of generic algebras

Let $A$ be a generic algebra, and let $\Pi$ be a $k$-dimensional affine subspace of $A$. An idempotent is an element $e\in A$ satisfying $e^2=e$. The idempotent bound. The affine subspace $\Pi$ contains at most $2^k$ distinct idempotents. This is suggested by the preceding analysis involving the proof of Proposition 2 and a square lemma; the claim is presented as plausible and no resolution is given here.