η-invariance of the local integrals of motion

Let ${\cal I}_n$, for $n=1,2,\ldots$, be the Local Integrals of Motion for the deformed $W$-algebra $W_{q,t}(\widehat{sl_N})$, defined by analytic continuation when necessary. Let $\eta$ denote the automorphism acting on these operators. η-invariance of the local integrals. The Local Integrals of Motion are $\eta$-invariant:

$$\eta({\cal I}_n)={\cal I}_n,\qquad (n=1,2,\ldots).$$

This is one of the stated properties of the commuting family of local integrals of motion; the supplied text does not provide a proof or resolution status for this assertion.