Conjecture on twists of symmetric-power L-values

Let $\varphi\in S_k(N,\omega)$ be a primitive form, let $\eta$ be a primitive Dirichlet character, and let $L_f(s,\operatorname{Sym}^n\varphi,\eta)$ denote the finite $L$-function twisted by $\eta$. Write $\mathfrak{g}(\eta)$ for the Gauss sum, and let $\sim$ mean equality up to an element of $\mathbb{Q}(\varphi)\mathbb{Q}(\eta)$. Twisting conjecture for symmetric-power $L$-functions. If $\eta$ is even, then the critical sets of $L_f(s,\operatorname{Sym}^n\varphi,\eta)$ and $L_f(s,\operatorname{Sym}^n\varphi)$ coincide, and at a critical integer $m$,

$$L_f(m,\operatorname{Sym}^n\varphi,\eta)\sim\mathfrak{g}(\eta)^{\lceil(n+1)/2\rceil}L_f(m,\operatorname{Sym}^n\varphi),$$

unless $n$ is even and $m$ is odd to the left of the center of symmetry, in which case the exponent is $n/2$. If $\eta$ is odd and $n$ is even, a critical $m$ for the twisted function has either $m+1$ or $m-1$ critical for the untwisted function; for $m$ to the right of the center,

$$L_f(m,\operatorname{Sym}^n\varphi,\eta)\sim((2\pi i)^{\mp}\mathfrak{g}(\eta))^{n/2+1}L_f(m\pm1,\operatorname{Sym}^n\varphi),$$

while to the left of the center the exponent is $n/2$. If $\eta$ is odd and $n$ is odd, the critical sets coincide; for $k\geq3$, a critical $m$ for the twisted function has either $m+1$ or $m-1$ critical for the untwisted function, and

$$L_f(m,\operatorname{Sym}^n\varphi,\eta)\sim((2\pi i)^{\mp}\mathfrak{g}(\eta))^{(n+1)/2}L_f(m\pm1,\operatorname{Sym}^n\varphi).$$

The conjecture predicts how twisting changes symmetric-power special values: even twists preserve critical points and introduce Gauss-sum factors, whereas odd twists can shift the relevant critical integer and introduce powers of $2\pi i$ as well. The provided material gives no resolution status.