Positive entropy conjecture for composite symplectic Dehn twists

Let $S_1$ and $S_2$ be Lagrangian $n$-spheres in an $A_2^n$-configuration, intersecting transversely at a single point, and let $\tau_1$ and $\tau_2$ be symplectic Dehn twists along them. For integers $k,\ell$ with $k\ell<0$, consider the composition

$$\tau=\tau_1^k\tau_2^\ell.$$

Positive entropy conjecture for composite symplectic Dehn twists. The measure-theoretic entropy of $\tau$ with respect to Lebesgue measure $\operatorname{vol}$ is positive:

$$h_{\operatorname{vol}}(\tau)>0.$$

The conjecture proposes chaotic measure-theoretic dynamics for compositions of Dehn twists supported on an $A_2^n$-configuration. The supplied text does not state a resolution.