Conjecture on the isolated-singularity injection for Kloosterman motives

Let $N(V_\lambda)$ and $E(V_\lambda)$ be endomorphisms of $V_\lambda$, induced from the endomorphisms $N$ and $E$ of the standard representation of $\mathrm{SL}_{n+1}$. For the injection

$$\mathrm{gr}^{W'}_{n|\lambda|+1} \mathrm{im}(\beta)^{G_\lambda\times \mu_{n+1},\chi_n}(-1)\hookrightarrow \mathrm{gr}^W_{n |\lambda|+1} \mathrm{H}^{n |\lambda|-1}_{\acute{e}t,c}(\mathcal{K}_{\bar{\mathbb{Q}}})^{G_\lambda\times \mu_{n+1},\chi_n} (-1) = (\mathrm{M}^\lambda_{n+1})_\ell,$$

Isolated-singularity conjecture. The morphism above is an isomorphism for any $n$ and $\lambda$ when the matrix $N(V_\lambda)+E(V_\lambda)$ is invertible. This would imply unramifiedness of $(\mathrm{M}^\lambda_{n+1})_\ell$ in the corresponding situation; the source gives no resolution of the conjecture.