Twisted ECH functoriality for capping and decoupling cobordisms

Let $(W,\omega)$ be a $\Sigma$-capping or $\Sigma$-decoupling cobordism from $(M_-,xi_-)$ to $(M_+,xi_+)$, and define the closed $2$-forms on the boundary by

$$\Omega_\pm=\omega|_{TM_\pm}.$$

For each boundary contact manifold, let $c(M_\pm,xi_\pm;\Omega_\pm)$ denote its twisted contact invariant in ${\operatorname{ECH}}(M_\pm,xi_\pm;0,\Omega_\pm)$, and let $U^k$ be the $k$-fold $U$-map.

Twisted ECH cobordism conjecture. If $c(M_+,xi_+;\Omega_+)$ vanishes, then $c(M_-,xi_-;\Omega_-)$ also vanishes. Moreover, if $c(M_+,xi_+;\Omega_+)$ lies in the image of $U^k$ on ${\operatorname{ECH}}(M_+,xi_+;0,\Omega_+)$ for some $k\in\mathbb N$, then $c(M_-,xi_-;\Omega_-)$ lies in the image of $U^k$ on ${\operatorname{ECH}}(M_-,xi_-;0,\Omega_-)$.

This conjectural extension of the exact-cobordism result would remove the exactness condition on $\omega$ for $\Sigma$-capping and $\Sigma$-decoupling cobordisms, using twisted coefficients to record the boundary restrictions of the symplectic form. The preceding discussion presents vanishing results as evidence, but the statement is not identified as resolved in the supplied text.