YES

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

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

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

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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