YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
F(H(x),y) -> F(H(x),I(I(y)))
F(x,G(y)) -> F(I(x),G(y))
I(x) -> x

CP(R):
F(H(___y125),I(I(G(___y128)))) -> F(I(H(___y125)),G(___y128))
F(I(H(___y134)),G(___y133)) -> F(H(___y134),I(I(G(___y133))))

critical-pair-closing system C:
I(x) -> x

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
I(x) -> x

CP(R):


critical-pair-closing system C:


CR(C): see below.

---- CR(R)
---- R is empty.