YES

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

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

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

CR(C): see below.

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