YES

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

CP(R):
b() -> c()
b() -> e()
c() -> b()
c() -> e()
e() -> b()
e() -> c()
a() -> e()
e() -> a()

critical-pair-closing system C:
a() -> e()
b() -> d()
c() -> a()
d() -> e()
h(x) -> e()

CR(C): see below.

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