Cohomological P-invariant conjecture for Hilbert modular forms

Let $\Sigma_{\infty}$ be the set of archimedean places, let $S\subset\Sigma_{\infty}$, and let $r\in\{0,1\}^{S}$. Let $\pi$, $\tau$, the cohomology classes $\mathbf{g}^{(\delta)}$, $\mathbf{h}^{(2)}$, the vector $Y_{+}^{(r)}\boldsymbol\phi$, and the periods $\nu^{\Sigma_b}(\tau)$ be as in the paper; let $\Sigma_{ub}$ denote the unbalanced places and let $\varepsilon=(\varepsilon_v)$ with $\varepsilon_v\equiv m+\kappa_0+1\pmod 2$ for every $v\in\Sigma_{ub}$. Cohomological P-invariant conjecture. For every $S\subset\Sigma_{\infty}$ and $r\in\{0,1\}^{S}$, there exists a complex number $P(\pi^\sigma,S,r)$ such that, for every $\sigma\in\operatorname{Aut}(\mathbb{C})$,

$$\left|\frac{\langle\mathbf{g}^{(\delta)},\mathbf{h}^{(2)}\otimes Y_{+}^{(r)}\boldsymbol\phi\rangle}{P(\pi,\Sigma_{ub},\varepsilon)\nu^{\Sigma_b}(\tau)}\right|^\sigma= \left|\frac{\langle(\mathbf{g}^{(\delta)})^\sigma,(\mathbf{h}^{(2)})^\sigma\otimes Y_{+}^{(r)}\boldsymbol\phi\rangle}{P(\pi^\sigma,\Sigma_{ub},\varepsilon)\nu^{\Sigma_b}(\tau^\sigma)}\right|.$$

The proposed quantities $P(\pi^\sigma,S,r)$ are intended as a cohomological interpretation of Shimura's $P$-invariant. The source presents this as an equivalent formulation in the central-value case, motivated by properties of Shimura's invariants, but gives no resolution.