The two distinguished-wave-determined slopes conjecture

Let $M$ be a tunnel-number-one orientable $3$-manifold with incompressible torus boundary. A tunnel of $M$ determines a distinguished-wave-determined slope on $\partial M$. The two-slope conjecture. The set of slopes determined by the tunnels of $M$ has cardinality at most two. This is motivated by the survey of tunnel-number-one manifolds with incompressible torus boundary; the supplied source gives no resolution, so the conjecture remains open.