YES

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

CP(R):


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

CR(C): see below.

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