YES

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

CP(R):
F(I(___y12),___y14) -> G(H(I(___y12)))

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

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
F(H(x),y) -> G(H(x))
H(I(x)) -> I(x)
F(I(x),y) -> G(I(x))
CP(R):
F(I(___y42),___y44) -> G(H(I(___y42)))