concatenation of context sensitive and context-free is context sensitive or not? Assume that $L_1$ is context sensitive language and $L_2$ is context free language, is the language $L_1 * L_2$ context-sensitive or not?

I almost sure that is not, but can't prove it.
Suppose that $L$ is a context-sensitive language which is not context-free. Then $L\{\epsilon\}$ is not context-free (here $\epsilon$ is the empty word), while $L\emptyset$ is context-free.