Is there any Hahn Banach type theorem for symmetric sublinear functionals (with additive assumptions) ?

The Hahn–Banach theorem states that: Given a sublinear functional $S: V \rightarrow \mathbb R$, if $T: U \rightarrow \mathbb R$ is a linear functional on a linear subspace $U \subseteq V$ that is dominated by $S$ on $U$, then there exists a linear extension of $T$ to $V$ that is dominated by $S$ on $V$.

Now, let us consider a symmetric multisublinear (positively homogeneous and subadditive in every component) continous functional $S: V\times\cdots\times V \rightarrow\mathbb R$ satisfying some good additionnal assumptions and a symmetric multilinear continous functional $T: U\times\cdots\times U \rightarrow \mathbb R$ that is dominated by $S$ on $U\times\cdots\times U$. Does there exist an extension of $T$ to $V\times\cdots\times V$ that is dominated by $S$ on $V\times\cdots\times V$?