YES

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

CP(R):
___y3 -> f(f(___y3))
f(f(___y5)) -> ___y5

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

CR(C): see below.

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