YES

Problem:
 f(x) -> g(x)
 f(x) -> h(f(x))
 h(f(x)) -> h(g(x))
 g(x) -> h(g(x))

Proof:
 Church Rosser Transformation Processor:
  strict:
   f(x) -> g(x)
   f(x) -> h(f(x))
   h(f(x)) -> h(g(x))
   g(x) -> h(g(x))
  weak:
   
  critical peaks: 4
   g(x) <-0|[]- f(x) -1|[]-> h(f(x))
   h(g(x)) <-0|0[]- h(f(x)) -2|[]-> h(g(x))
   h(f(x)) <-1|[]- f(x) -0|[]-> g(x)
   h(h(f(x))) <-1|0[]- h(f(x)) -2|[]-> h(g(x))
  Closedness Processor (*strongly -- <=7 steps*):
   strict:
    
   weak:
    
   Qed