YES

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

CP(R):
b() -> f(a())
f(a()) -> b()

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

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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