The integration conjecture for anomaly-free coefficient systems on closed stratified surfaces

Let $\Sigma$ be a closed stratified surface and let $A$ be an anomaly-free coefficient system in $\mathrm{Cat}_{\EuScript{E}}^{\mathrm{fs}}$ on $\Sigma$. Here $\int_{\Sigma} A$ denotes the integration of $A$, and $(\EuScript{E},u_{\Sigma})$ denotes the coefficient category together with an object $u_{\Sigma}$ of $\EuScript{E}$.

Integration conjecture. For every such $\Sigma$ and $A$, there is an equivalence

$$\int_{\Sigma} A \simeq (\EuScript{E},u_{\Sigma}),$$

where $u_{\Sigma}$ is an object in $\EuScript{E}$.

This asserts that integration over any closed stratified surface with anomaly-free coefficients produces the coefficient category $\EuScript{E}$ together with a distinguished object. The supplied text does not state whether the claim has been proved or remains open.