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(___y5),I(I(G(___y8)))) -> F(I(H(___y5)),G(___y8))
F(I(H(___y14)),G(___y13)) -> F(H(___y14),I(I(G(___y13))))

critical-pair-closing system C:
F(H(x),y) -> F(H(x),I(I(y)))
I(x) -> x

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
F(H(x),y) -> F(H(x),I(I(y)))
I(x) -> x
CP(R):