YES

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

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

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

CR(C): see below.

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