Let $V$ be a $\mathbb{C}$-vector space of dimension $N$, let $d$ be a positive integer, let $l \leq N$ be a positive integer, and let $U \subseteq S^d(V)$ be a linear subspace of codimension $k=\binom{l+d-1}{d}$. Does there exist a linear subspace $W \subseteq V$ of dimension $l$ for which $S^d(V)=U \oplus S^d(W)$?

I tried to prove yes by showing that the dimension of the subvariety of $Gr(k,S^d(V))$, consisting of subspaces of the form $S^d(W)$ for an $l$-dimensional linear subspace $W \subseteq V$, is greater than the dimension of the subvariety of subspaces that intersect $U$ non-trivially, but this is emphatically not the case. The former has dimension $l(N-l)$, while the latter has dimension $k(\binom{N+d-1}{d}-k)$. (See here)