Let $A,B$ be subfactors of a II$_1$ factor $M$ with $A*B\simeq M$. That is, $A$ and $B$ are freely independent with respect to the trace and $M\simeq A\vee B$. We'll call $B$ a free complement for $A$ in $M$.

My first question is simple (to state): if a free complement exists is it necessarily unique? My intuition says the answer should be ''no'', but I'm not aware of any simple examples?

Moving from the general to the specific, I'm really interested in the following example of Voiculescu. Let $H$ be a real, infinite dimensional Hilbert space and $\Phi(H_{\mathbb{C}})=\mathbb{C}\Omega\oplus\bigoplus_{n=1}^\infty{H^{\otimes n}_{\mathbb{C}}}$ the full Fock space over its complexification. If $\ell(f)$ is the left creation operator defined by $\ell(f)\Omega=f$, $\ell(f)\xi=f\otimes\xi$ when $\xi\perp\Omega$ and $\mathrm{h}$ is a subspace of $H$ then consider the von Neumann algebra $$L(\mathrm{h})=\{s(f):~f\in\mathrm{h}\}''\qquad\hbox{where}\qquad s(f)=\frac{\ell(f)+\ell(f)^*}{2}.$$ $L(H)$ is isomorphic to the free group factor $L(F_\infty)$ acting standardly with cyclic trace vector $\Omega$. If $\mathrm{h}\oplus\mathrm{k}=H$ then $L(\mathrm{h})*L(\mathrm{k})\simeq L(H)$. The second question is a special case of the first: is $L(\mathrm{k})$ the only free complement to $L(\mathrm{h})$ in $L(H)$?