Weak symplectic cobordism conjecture for monopole Floer contact elements

Let $(W,\omega)$ be a weakly symplectic cobordism from $(Y_1,\xi_1)$ to $(Y_2,\xi_2)$. For $i=1,2$, let $\eta_i\subset Y_i$ be a $1$-cycle dual to $\omega|_{Y_i}$, and let $nu\subset W$ be a $2$-cycle with

$$\partialnu=-\eta_1\cupeta_2.$$

Write $s_{\omega}$ for the canonical spin$^c$ structure associated to $\omega$, $s_{xi_i}$ for the contact spin$^c$ structures, and $\Gamma_{nu}$ and $\Gamma_{-\eta_i}$ for the corresponding local coefficient systems.

Weak symplectic cobordism conjecture. The map

$$\widecheck{HM}(W,\mathfrak{s_{\omega}};\Gamma_{\nu}):\widecheck{HM}(-Y_2,\mathfrak{s}_{\xi_2};\Gamma_{-\eta_2})ra\widecheck{HM}(-Y_1,\mathfrak{s}_{\xi_1};\Gamma_{-\eta_1})$$

preserves the contact elements:

$$\widecheck{HM}(W,\mathfrak{s_{\omega}};\Gamma_{\nu})(\phi_{\xi_2})\doteq\phi_{\xi_1}.$$

Exact and strong symplectic cobordisms are known to preserve contact elements, while the weakly symplectic case described here is the proposed extension and remains open.