YES

   1: f(x1) -> g(x1)
   2: f(x1) -> h(f(x1))
   4: g(x1) -> h(g(x1))

@Development Closedness
--- R
   1: f(x1) -> g(x1)
   2: f(x1) -> h(f(x1))
   4: g(x1) -> h(g(x1))

--- S
   1: f(x1) -> g(x1)
   2: f(x1) -> h(f(x1))
   4: g(x1) -> h(g(x1))


NOTE: input TRS is reduced

original is
   1: f(x1) -> g(x1)
   2: f(x1) -> h(f(x1))
   3: h(f(x1)) -> h(g(x1))
   4: g(x1) -> h(g(x1))

reduced to
   1: f(x1) -> g(x1)
   2: f(x1) -> h(f(x1))
   4: g(x1) -> h(g(x1))