Specialized commutativity conjecture for the nonlocal integrals of motion

At the specialization $s=2$, let ${\cal G}_n^{DV}$ denote the nonlocal integrals of motion for the deformed Virasoro algebra, defined by the multiple contour integrals in the source. Let $\eta$ denote the Dynkin automorphism. Specialized nonlocal commutativity conjecture. Upon specialization $s=2$,

$$[{cal G}_n^{DV},{\cal G}_m^{DV}]=0,\qquad \eta({\cal G}_n^{DV})={\cal G}_n^{DV},\qquad (m,n=1,2,\ldots).$$

This asserts mutual commutativity and Dynkin-automorphism invariance for the specialized nonlocal integrals. The supplied text gives no resolution status.