Archimedean rationality conjecture for cup products of automorphic forms

Let $f$ and $f'$ be the forms specified in Theorem BS, chosen so that for every $v\in S_\infty$ their local components lie in the indicated coefficient-field-rational subspaces of the minimal $K_{H,v}$-types. Let $I^*_v(f_v,f'_v)$ denote the relevant archimedean cup-product pairing. Archimedean rationality conjecture. For every $v\in S_\infty$,

$$I^*_v(f_v,f'_v)\in E(\pi(q))\cdot E(\pi'(q)).$$

This is proposed as an archimedean analogue of a known local rationality lemma and concerns the arithmetic nature of the archimedean pairing; no resolution is supplied in the source.