YES

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

CP(R):
F(___y13) -> G(___y13)

critical-pair-closing system C:
F(c(x)) -> G(x)
G(x) -> F(x)
c(x) -> x

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
F(c(x)) -> G(x)
G(x) -> F(x)
c(x) -> x
CP(R):
F(___y31) -> G(___y31)