Twisted symplectic cohomology vanishing for Higgs branch moduli spaces with exact Lagrangians

Let $Y$ be a Higgs branch moduli space, meaning a hyperkähler reduction of a unitary action on a flat vector space, and let $L\subset (Y,\omega_J)$ be a closed exact Lagrangian. Write $\omega_I$ for the other Kähler form in the hyperkähler structure, and let $\underline{\Lambda}_{\tau\omega_I}$ denote the coefficient system twisted by the transgression of $\omega_I$.

Twisted symplectic cohomology vanishing conjecture. A deformation argument should imply

$$SH^*(Y,\omega_J,\underline{\Lambda}_{\tau\omega_I})=0.$$

Consequently, $H^2(L;\mathbb R)\neq 0$, $\pi_2(L)$ is infinite, and the Hofer–Zehnder capacity of $(Y,\omega_J)$ is finite and bounded above by the capacity bound in the source; in particular, there are no Lagrangian spheres when $\dim_{\mathbb C}Y>2$.

The proposed vanishing would extend deformation and twisted-symplectic-cohomology methods used for noncompact symplectic manifolds and would yield geometric restrictions on exact Lagrangians in Higgs branch moduli spaces. The status of the implication is not resolved in the supplied text.