Generation of the full Hecke property from the $GL_n$ Hecke property

Let $E$ be an arbitrary $n$-dimensional local system on $X$. Let ${\mathcal S}_E\in D(\Bun_n)$ have the $GL_n$-Hecke eigen-property with respect to $E$, meaning that the relevant Hecke eigen-isomorphism is given only for the tautological representation $V_0$ and satisfies the stated compatibility condition.

Full Hecke-property conjecture. Then ${\mathcal S}_E$ satisfies the full Hecke property.

This claim asserts that, for $GL_n$, the eigen-condition for the tautological representation suffices to recover all Hecke eigen-isomorphisms. The source gives no evidence that the assertion has been resolved.