Left-handed stabilization has vanishing contact homology

Let $(M,\alpha)$ be a cooriented contact manifold of dimension at least $3$. Let $(\Sigma,\psi)$ be a supporting open book for $(M,\alpha)$, and suppose that $\Sigma$ contains a Lagrangian ball $L$ whose boundary is a Legendrian sphere in $\partial\Sigma$.

Left-handed stabilization conjecture. The left-handed stabilization of $(M,\alpha)$ along $L$ has vanishing contact homology:

$$HC_*\bigl(\operatorname{Stab}_{L}^{-}(M,\alpha)\bigr)=0.$$

The paper states this claim among its conjectures, although its introduction also describes a theorem proving algebraic overtwistedness for a boundary-parallel Lagrangian ball in a compatible open book. The precise relationship between that theorem and this formulation should be checked.