YES

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

CP(R):
f(h(g(___y7750))) -> a()
f(g(h(___y7749))) -> a()
f(h(h(___y7752))) -> b()
f(h(h(___y7753))) -> b()

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

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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