Reduction of Entov–Polterovich quasimorphisms to Donaldson divisors

Let $(X,\Sigma)$ be a pair consisting of a closed monotone symplectic manifold and a Donaldson divisor. Let

$$\mu_X^{EP}:\widetilde{\operatorname{Ham}}(X)\longrightarrow\mathbb{R}$$

be an Entov–Polterovich quasimorphism such that the skeleton $\Delta$ is small. Let $\Theta$ denote the reduction map associated with the Donaldson divisor.

Reduction conjecture. The reduction of $\mu_X^{EP}$ to $\Sigma$ is also an Entov–Polterovich quasimorphism: there exists an Entov–Polterovich quasimorphism

$$\mu_\Sigma^{EP}:\widetilde{\operatorname{Ham}}(\Sigma)\longrightarrow\mathbb{R}$$

that satisfies

$$\Theta^*\mu_X^{EP}=\mu_\Sigma^{EP}.$$

This conjecture concerns the expected compatibility between Entov–Polterovich quasimorphisms on a monotone symplectic manifold and on a Donaldson divisor. The source presents it as an expected consequence related to the reduction of quasimorphisms; no resolution is supplied here.