Handle-decomposition independence of the sutured monopole Floer contact map

Let $(M,\Gamma,\xi_M)$ be a sutured contact manifold contained in $(M',\Gamma')$, and let $\xi$ be a contact structure on $M'\smallsetminus \operatorname{int}(M)$ with dividing set $\Gamma\cup \Gamma'$. For a contact handle decomposition $H$ of $(M'\smallsetminus \operatorname{int}(M),\Gamma\cup \Gamma',\xi)$, let

$$\Phi_{\xi,H}:\underline{SHM}(-M,-\Gamma)\longrightarrow\underline{SHM}(-M',-\Gamma')$$

be the corresponding composition of contact handle attachment maps. Handle-decomposition independence conjecture. The map $\Phi_{\xi,H}$ is independent of the handle decomposition $H$. This would make the monopole Floer analogue of the sutured Floer contact cobordism map canonical, rather than dependent on auxiliary handle-decomposition choices. The source does not state a resolution.