YES

---- CR(R)
---- Hot decreasingness.
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()

C:
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()

Label: