YES

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

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

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

CR(C): see below.

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