YES

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

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

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

CR(C): see below.

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