YES

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

CP(R):
g(b()) -> f(g(a()))

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

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

---- CR(R)
---- R is empty.