About the closedness of a certain product of two closed operators?

Question

Let $H$ be a Hilbert space. Let $S∈B(H)$ and let $T$ be a densely defined closed operator such that $TS\subset ST$. Assume further that $T$ is boundedly invertible.

Is it true that $ST$ is closed?

In my problem, $S$ and $T$ are both self adjoint and positive.