Existence of product vectors and their partial conjugates in a pair of spaces

Let $D$ and $E$ be subspaces of the tensor product of the $m$ and $n$ dimensional complex spaces, with codimensions $k$ and \ell$, respectively. We show that if $k+\mathrm{\ell }<m+n-2$ then there must exist a product vector in $D$ whose partial conjugate lies in $E$. If $k+\mathrm{\ell }>m+n-2$ then there may not exist such a product vector. If $k+\mathrm{\ell }=m+n-2$ then both cases may occur depending on $k$ and $\mathrm{\ell }$.