YES

1 decompositions
 #0 -----------
   1: F(H(x),y) -> F(H(x),I(I(y)))
   2: F(x,G(y)) -> F(I(x),G(y))
   3: I(x) -> x

@Strongly Commuting
--- R
   1: F(H(x),y) -> F(H(x),I(I(y)))
   2: F(x,G(y)) -> F(I(x),G(y))
   3: I(x) -> x

--- S
   1: F(H(x),y) -> F(H(x),I(I(y)))
   2: F(x,G(y)) -> F(I(x),G(y))
   3: I(x) -> x