Handle-decomposition independence conjecture for sutured instanton cobordism maps

Let $(M,\Gamma)$ be a sutured submanifold of $(M',\Gamma')$, and let $\xi$ be a contact structure on $M'\setminus\operatorname{int}(M)$ with dividing set $\Gamma\cup\Gamma'$. Given a contact-handle decomposition $H$ of this cobordism, let

$$\Phi_{\xi,H}:\underline{\operatorname{SHI}}(-M,-\Gamma)\to\underline{\operatorname{SHI}}(-M',-\Gamma')$$

be the corresponding composition of contact handle attachment maps. Handle-independence conjecture. The map $\Phi_{\xi,H}$ is independent of $H$. This would make the map depend only on the contact structure $\xi$, as in the corresponding constructions in sutured Heegaard Floer and monopole homology.