The period-integral commutator relation

Assume the setup denoted by $\thref{commsetup}$, and let $I=\{1,2\}$ and $[I]=\{1\}$. Let $c_{V\boxtimes W,\ftwo}$, $c_{W\boxtimes V,\ftwo}$, and $c_{V\otimes W,\fone}$ be the elements defined via the period-push construction.

Period-integral commutator relation. Under $\thref{commsetup}$,

$$c_{V\boxtimes W,\ftwo}-c_{W\boxtimes V,\ftwo}=\coev(c_{V\otimes W,\fone})\in\Cor_{\hk^{\glob}_{G,\ftwo},\IC_{V\boxtimes W}}(\cP_X\boxtimes\uk_{C^2},\cP_X\boxtimes\uk_{C^2}\langle d_V+d_W\rangle).$$

The relation reformulates the automorphic commutator identity using the period-push construction and the coevaluation map. Its status is not specified in the source.