Generalization of a standard algebraic group theory result for a tensor problem

$$\DeclareMathOperator\GL{GL}$$Let $$X$$, $$Y$$, $$Z$$ be $$\mathbb{C}$$-vector spaces, and let $$A\subseteq X$$ and $$B\subseteq Y$$ and $$C\subseteq Z$$ be linear subspaces. Let $$V=X \otimes Y \otimes Z$$, acted on algebraically by $$G\mathrel{:=}\GL(X)\times \GL(Y)\times \GL(Z)$$ in the standard way. Let $$W=A \otimes B \otimes C \subseteq V$$.

For each $$l \in \mathbb{N}$$, define \begin{align}\tag{1}\label{1} S_l=\{v \in V : \dim(\overline{G \cdot v} \cap W)\leq l\}. \end{align}

Is it true that $$S_l \subseteq V$$ is Zariski closed for all $$l \in \mathbb{N}$$? If this statement is true, I would prefer a proof that uses classical algebraic group theory/algebraic geometry, as I am not very familiar with sheaves and schemes.

Note that if $$W=V$$, then this becomes a very standard result in algebraic group theory (see e.g. Lemma 1.4 in Brion - Introduction to actions of algebraic groups).