Let $V$ be a $\mathbb{C}$-vector space of dimension $N \geq 2$, let $d$ be a positive integer, let $l < N$ be a positive integer, and let $U \subseteq S^d(V)$ be a linear subspace of codimension $k=\binom{l+d-1}{d}$. Suppose that there exists a basis of $U$ of the form $\{u_1^{\vee d},\dots, u_r^{\vee d}\}$ such that $$ \text{span}\{u_1^{\vee d-1},\dots, u_r^{\vee d-1}\}=S^{d-1}(V), $$ where $r=\binom{N+d-1}{d}-k$. Does there exist a linear subspace $W \subseteq V$ of dimension $l$ for which $S^d(V)=U \oplus S^d(W)$?

Here, $\vee$ denotes the symmetric product.

This is a variant of my previous question, which did not assume any particular form for $U$.

Let $V$ be a $\mathbb{C}$-vector space of dimension $N \geq 2$, let $d$ be a positive integer, let $l < N$ be a positive integer, and let $U \subseteq S^d(V)$ be a linear subspace of codimension $k=\binom{l+d-1}{d}$. Suppose that there exists a basis of $U$ of the form $\{u_1^{\vee d},\dotsc, u_r^{\vee d}\}$ such that $$ \operatorname{span}\{u_1^{\vee d-1},\dotsc, u_r^{\vee d-1}\}=S^{d-1}(V), $$ where $r=\binom{N+d-1}{d}-k$. Does there exist a linear subspace $W \subseteq V$ of dimension $l$ for which $S^d(V)=U \oplus S^d(W)$?

Here, $\vee$ denotes the symmetric product.

This is a variant of my previous question Given a subspace $U \subseteq S^d(V)$, does there always exist a complement of the form $S^d(W)$?, which did not assume any particular form for $U$.