Let $G$ be a reductive algebraic group over $\mathbb{C}$, and let $V$ be a finite-dimensional (rational) representation of $G$. Then $G$ acts on $V$ as a space via this representation. Hence $G$ acts on the coordinate ring of $V$, $\mathbb{C}[V]=\text{Sym}^\bullet(V^*)$.
We will say that $V$ is $\mathit{multiplicity}$-$\mathit{free}$ if $\mathbb{C}[V]$ is multiplicity-free (i.e. no irrep shows up more than once) as a representation of $G$. Note that this is equivalent to $V$ having an open $B$-orbit for a Borel subgroup $B\subseteq G$ (i.e. $V$ is a $\mathit{spherical}$ $G$-variety).
My question is: suppose that $V$ and $W$ are representations of $G$ and that $V\oplus W$ is multiplicity-free. Then is it true that $V\oplus W^*$ is multiplicity-free?
My feeling is that the answer is yes, but I don't see how to prove this either on the representation-theoretic level or by trying to show there is an open $B$-orbit.