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(___y49) -> G(___y49)

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

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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