Deligne's algebraicity conjecture for twisted symmetric fifth L-functions

Let $f(\tau)=\sum_{n=1}^{\infty}a_f(n)q^n\in S_\kappa(N,\omega)$ be a normalized elliptic newform of weight $\kappa\geq 2$, and let $\chi$ be a Dirichlet character. For a critical point $m$ of ${\rm Sym}^5(f)$, let $G(\chi)$ denote the Gauss sum, let $c^\pm({\rm Sym}^5(f))$ be Deligne's periods, and for $\sigma\in{\rm Aut}(\mathbb C)$ let ${}^\sigma\!f$ and ${}^\sigma\!\chi$ denote the conjugates under $\sigma$. Deligne's conjecture. For every $\sigma\in{\rm Aut}(\mathbb C)$,

$$\sigma\left(\frac{L(m,{\rm Sym}^5(f)\otimes\chi)}{(2\pi\sqrt{-1})^{3m}\cdot G(\chi)^3\cdot c^\pm({\rm Sym}^5(f))}\right)=\frac{L(m,{\rm Sym}^5({}^\sigma\!f)\otimes{}^\sigma\!\chi)}{(2\pi\sqrt{-1})^{3m}\cdot G({}^\sigma\!\chi)^3\cdot c^\pm({\rm Sym}^5({}^\sigma\!f))},$$

where $\pm=(-1)^m\chi(-1)$. Here the critical points are the integers satisfying $2\kappa-1\leq m\leq3\kappa-3$. This is the special case of Deligne's conjecture concerning the algebraicity and Galois equivariance of critical values of twisted symmetric fifth $L$-functions; the source records no resolution status, so it remains open.