If the dimension of the null space of $T,S$ is $5$ and $3$ . What is the dimension of the null space of $T\circ S$?

Question

Let $V$ be a vector space with $\dim V =12$. Then $T$ and $S$ are linear operators on $V$.

We have that the dimension of the null space of $T$ is $5$ and the dimension of the null space of $S$ is 3.

Then what is the dimension of the null space of $T\circ S$?

I have found a post here Dimension of Range and Null Space of Composition of Two Linear Maps

By this post, I found that $\dim null(T\circ S)\leq \dim null(T)+\dim null(S)=5+3=8$.

Also $\dim range(T\circ S)\leq \min\{\dim range(T),\dim range(S)\}=\min\{7,9\}=7$

Hence $5\leq\dim null(T \circ S)\leq8$.

Is this correct?