YES

   1: f(g(g(x1))) -> a()
   2: f(g(h(x1))) -> b()
   3: f(h(g(x1))) -> b()
   4: f(h(h(x1))) -> c()
   5: g(x1) -> h(x1)
   6: a() -> b()
   7: b() -> c()

@Jouannaud and Kirchner's criterion
--- R
   1: f(g(g(x1))) -> a()
   2: f(g(h(x1))) -> b()
   3: f(h(g(x1))) -> b()
   4: f(h(h(x1))) -> c()
   5: g(x1) -> h(x1)
   6: a() -> b()
   7: b() -> c()

--- S
   1: f(g(g(x1))) -> a()
   2: f(g(h(x1))) -> b()
   3: f(h(g(x1))) -> b()
   4: f(h(h(x1))) -> c()
   5: g(x1) -> h(x1)
   6: a() -> b()
   7: b() -> c()