YES

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

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

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

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
a() -> b()
f(b()) -> g(b())
CP(R):