Let us consider the algebra $\mathfrak{gl}_{\infty}$ (or $\mathfrak{gl}_n$, or just any finite-dimensional semisimple Lie algebra; howerer, I am primarily concerned with the case of $\mathfrak{gl}_{\infty}$ and $\mathfrak{gl}_n$). I call its highest weight representation $V$ of weight $\lambda$ anti-dominant if $\lambda=\sum -k_i\omega_i$ where $\omega_i=(\dots 1\ 1\ 0\ 0\dots)$ is the $i$-th fundamental weight, $k_i\in\mathbb{Z}_{\geq 0}$ and only a finite number of them is nonzero.

Is it true that the tensor product of two irreducible anti-dominant representaions is semisimple? Or maybe at least the sub-representation generated by the highest weight vector of the tensor product is irreducible? How can one prove (disprove) this?