Spectral naturality under quasi-containment

Let $G$ be a locally compact group, and let $\tau_\rho$ denote the representation associated with a representation $\rho$. A representation $\rho$ is spectrally natural when $\Phi_{\mathrm{A}(\rho)}=\Phi_{\mathrm{A}(\rho)}$. Write $\sigma\leq_q\pi$ when $\sigma$ is quasi-contained in $\pi$, and let $\lambda$ be the left regular representation. Spectral naturality conjecture. (i) If $\sigma\leq_q\pi$ and $\pi$ is spectrally natural, then $\sigma$ is spectrally natural. (ii) If $\sigma\leq_q\lambda$, then $\sigma$ is spectrally natural. Part (ii) is presented as potentially useful for the examples developed later, but the source supplies no resolution of either assertion.