Conjectural period relations for symmetric-power liftings

Let $\mathit\Pi$ be the cohomological cuspidal automorphic representation under consideration, with central character $\omega_\mathit\Pi$, Whittaker vector $f_\mathit\Pi$, period $p(\mathit\Pi,\pm)$, and symmetric-power liftings ${\rm Sym}^n\mathit\Pi$ equipped with periods $p({\rm Sym}^n\mathit\Pi,\pm)$ for odd $n$ and $p({\rm Sym}^n\mathit\Pi)$ for even $n$. Conjectural period relations. For every $n\geq1$, there is a constant $C_{n,\infty}\in\mathbb C^\times$, unique up to $\mathbb Q^\times$ and depending only on $n$ and $\mathit\Pi_\infty$, such that the stated Galois-equivariance relations hold: for $n=2r+1$,

\sigma\left(\frac{p({\rm Sym}^{2r+1}\mathit\Pi},\pm)}{C_{2r+1,\infty}G(\omega_\mathit\Pi)^{2r(r+1)^2}p(\mathit\Pi,\pm)^{r+1}\lVert f_\mathit\Pi\rVert^{2r(r+1)(2r+1)/3}}\right)=\frac{p({\rm Sym}^{2r+1}{}^\sigma\!\mathit\Pi},\pm)}{C_{2r+1,\infty}G({}^\sigma\!\omega_\mathit\Pi)^{2r(r+1)^2}p({}^\sigma\!\mathit\Pi,\pm)^{r+1}\lVert f_{{}^\sigma\!\mathit\Pi}\rVert^{2r(r+1)(2r+1)/3}},

and for $n=2r$,

$$\sigma\left(\frac{p({\rm Sym}^{2r}\mathit\Pi)}{C_{2r,\infty}G(\omega_\mathit\Pi)^{(2r+1)r^2}\lVert f_\mathit\Pi\rVert^{r(r+1)(2r+1)/3}}\right)=\frac{p({\rm Sym}^{2r}{}^\sigma\!\mathit\Pi)}{C_{2r,\infty}G({}^\sigma\!\omega_\mathit\Pi)^{(2r+1)r^2}\lVert f_{{}^\sigma\!\mathit\Pi}\rVert^{r(r+1)(2r+1)/3}}.$$

These relations are proposed as period analogues of Deligne's conjecture for critical standard $L$-values of symmetric-power liftings; the source gives no resolution evidence, so the relations remain conjectural.