Trace and cotrace conjecture for sutured monopole Floer homology

Let $(M,\gamma)$ be a balanced sutured manifold, and let $\mathcal{W}=(W=M\times[0,1],Z=\partial M\times[0,1],[\xi])$ be the trace sutured cobordism from $(M\sqcup(-M),\gamma\cup(-\gamma))$, with $\xi$ a $[0,1]$-invariant contact structure on $Z$. Let $\mathcal{R}$ be the coefficient ring, let

$$i:\underline{\operatorname{SHM}}(M\sqcup(-M),\gamma\cup\gamma)\longrightarrow\underline{\operatorname{SHM}}(M,\gamma)\otimes\underline{\operatorname{SHM}}(-M,\gamma)$$

be the map from the Künneth formula, and define

$$\operatorname{tr}:\underline{\operatorname{SHM}}(M,\gamma)\otimes\underline{\operatorname{SHM}}(-M,\gamma)\longrightarrow\mathcal{R},\qquad \operatorname{tr}(a\otimes b)=b(a).$$

Here $\underline{\operatorname{SHM}}(-M,\gamma)$ is the dual of $\underline{\operatorname{SHM}}(M,\gamma)$. Trace and cotrace conjecture. With these settings,

$$\underline{\operatorname{SHM}}(\mathcal{W})=\operatorname{tr}\circ i.$$

The conjecture identifies the cobordism map of the trace cobordism with the canonical evaluation map after the Künneth map. The source gives no evidence of a resolution, so the status remains open.