YES

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

CP(R):
f(f(___y59)) -> g(___y59)
g(___y67) -> f(f(___y67))

critical-pair-closing system C:
f(x) -> f(f(x))
g(x) -> f(x)

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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