The functional-equation conjecture for normalized periods

Let $G$ be a reductive group and let $V\in{\operatorname{Rep}}(G)$, with $V^\vee$ its dual representation. Let ${\mathcal{P}}_V^{\operatorname{norm}}$ denote the normalized period associated with $V$. Functional-equation conjecture for normalized periods. For every $V\in{\operatorname{Rep}}(G)$, there is an isomorphism

$${\mathcal{P}}_V^{\operatorname{norm}} \cong {\mathcal{P}}_{V^\vee}^{\operatorname{norm}}.$$

This is expected to follow from a refinement of Fourier transforms on Banach–Colmez spaces; the one-dimensional case is verified in the paper, while the general vectorial case remains open.