Tightness characterization of efficiency for irreducible handlebody automorphisms

Let $H$ be a handlebody and let $f\colon H\to H$ be an irreducible automorphism with associated $f$-invariant measured laminations $(\Lambda,\mu)$ and $(\Omega,\nu)$. The pair $((\Lambda,\mu),(\Omega,\nu))$ is tight if, for every leaf $L\in\Lambda$ and every simple closed curve $\gamma\subseteq L$, the disc $\Delta'\subseteq L$ bounded by $\gamma$ satisfies

$$\nu(\Delta')\leq\nu(\Delta)$$

for every disc $\Delta\subseteq H$ transverse to $\Omega$ with $\partial\Delta=\gamma$.

Tightness conjecture. The automorphism $f$ is efficient if and only if its associated pair of $f$-invariant laminations is tight.

Tightness is stronger than the incompressibility property and weaker than the no-backtracking condition; it is already known to imply efficiency. The conjecture asserts that tight invariant laminations can be realized for every mapping class, which would establish the converse implication.