Independence-of-$\ell$ conjecture for the spectral Bernstein center map

Let $G$ be a reductive group over the local field $E$, let $W_E$ be its Weil group, and let $\mathcal Z^{\mathrm{spec}}(G,\mathbb Q)$ be the natural rational form of the spectral Bernstein center. For every prime $\ell\neq p$, write $\mathbb Q_\ell(\sqrt q)$ for the corresponding coefficient field. Independence-of-$\ell$ conjecture. There is a necessarily unique map

$$\mathcal Z^{\mathrm{spec}}(G,\mathbb Q(\sqrt q))\to\mathcal Z(G(E),\mathbb Q(\sqrt q))$$

whose base change to every $\mathbb Q_\ell$ with $\ell\neq p$ is the composite

$$\mathcal Z^{\mathrm{spec}}(G,\mathbb Q_\ell(\sqrt q))\to\mathcal Z^{\mathrm{geom}}(G,\mathbb Q_\ell(\sqrt q))\to\mathcal Z(G(E),\mathbb Q_\ell(\sqrt q)).$$

This would make the constructed $L$-parameters independent of $\ell$ in the relevant sense; related conjectures concern the stable Bernstein center and the image of this map.