The semi-local period-integral commutator relation

Assume the general assumption denoted by $\thref{genassum}$. Let $c_{V\boxtimes W,\ftwo}^{\sloc}$, $c_{W\boxtimes V,\ftwo}^{\sloc}$, and $c_{V\otimes W,\fone}^{\sloc}$ be the elements defined via the semi-local composition construction.

Semi-local period-integral commutator relation. Under $\thref{genassum}$,

$$c_{V\boxtimes W,\ftwo}^{\sloc}-c_{W\boxtimes V,\ftwo}^{\sloc}=\coev_{\sloc}(c_{V\otimes W,\fone}^{\sloc})\in\Cor_{\hk^{\Xp,\sloc}_{G,\ftwo},\IC_{V\boxtimes W}}(\d^X_{\ftwo,\sloc},\d^X_{\ftwo,\sloc}\langle d_V+d_W\rangle).$$

This is the semi-local, or primitive, variant of the period-integral commutator relation. The source gives no evidence of a resolution.