Composition of a bounded Lipschitz continuous function with a Sobolev function

Question

Let $\Omega$ be a bounded open subset of $\mathbb{R}^2$ supposed convex with sufficiently smooth boundary. Let $T$ be a compactly supported Lipschitz continuous function that is extended to the whole $\mathbb{R}^2$ with zero. Let $\varphi \in H^1(\Omega,\mathbb{R}^2)$. Is $T \circ \varphi \in L^{\infty}(\Omega)$? If yes, is it possible to relate its $L^{\infty}$-norm with the one of $T$? I do thank you very much. Best wishes.