Maximality conjecture for a family of $p^k$-central spaces

Let $p$ be a prime, let $d=p^k$, and let $A=F[x,y]$ be a cyclic algebra of degree $d$. For some $0\leq e\leq k-1$, set

$$V=F[x^{p^e}]y+F[y^{p^{k-e}}]x.$$

This space is $p^k$-central: every element has its $p^k$th power in $F$.

Maximality conjecture. The $p^k$-central space $V$ is maximal with respect to inclusion.

The preceding proposition establishes that $V$ is $p^k$-central, while the supplied text does not prove its maximality or provide a counterexample. Thus the conjectural part concerns maximality among $p^k$-central subspaces.