The Wronskian power criterion for bases of top powers

Let $U\in\mathcal{G}_k\mathcal{P}_n$ admit a basis of top powers, and let $W(U)$ denote its Wronskian covariant. The notation $U^{(\ell-1)}$ denotes the associated derived subspace, and $W(U^{(\ell-1)})$ its Wronskian. Suppose that

$$W(U)=W\big(U^{(\ell-1)}\big)^\ell,$$

where the equality is relative to $\mathcal{G}_1\mathcal{P}_{k\ell}$, that is, up to a non-zero scalar multiple. Wronskian power criterion. If $W(U^{(\ell-1)})$ has $k$ distinct finite, simple roots, then $U$ admits a basis of top powers. This gives a proposed converse to the preceding necessary Wronskian power condition; its status is not determined by the supplied text.