YES

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

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

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

CR(C): see below.

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