YES

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

Proof:
 Church Rosser Transformation Processor:
  strict:
   
  weak:
   
  critical peaks: 2
   g(f(x)) <-0|[]- f(x) -2|[]-> h(f(x))
   h(f(x)) <-2|[]- f(x) -0|[]-> g(f(x))
  Shift Processor lab=left (dd) (force):
   strict:
    f(x) -> g(f(x))
    f(x) -> h(f(x))
    g(f(x)) -> p(p(h(f(x))))
    h(f(x)) -> p(h(f(x)))
    p(h(f(x))) -> p(p(h(f(x))))
    f(x) -> h(f(x))
    f(x) -> g(f(x))
    h(f(x)) -> p(h(f(x)))
    p(h(f(x))) -> p(p(h(f(x))))
    g(f(x)) -> p(p(h(f(x))))
   weak:
    f(x) -> g(f(x))
    h(x) -> p(h(x))
    f(x) -> h(f(x))
    g(x) -> p(p(h(x)))
   Shift Processor (ldh) lab=left (force):
    strict:
     
    weak:
     
    Rule Labeling Processor:
     strict:
      
     weak:
      
     rule labeling (right):
      f(x) -> g(f(x)): 1
      h(x) -> p(h(x)): 0
      f(x) -> h(f(x)): 1
      g(x) -> p(p(h(x))): 0
     Rule Labeling Processor:
      strict:
       
      weak:
       
      rule labeling (left):
       f(x) -> g(f(x)): 0
       h(x) -> p(h(x)): 0
       f(x) -> h(f(x)): 0
       g(x) -> p(p(h(x))): 0
      Decreasing Processor:
       The following diagrams are decreasing:
       peak:
        g(f(x)) <-0|[==0,==1,==1]- f(x) -2|[==0,==1,==1]-> h(f(x))
       joins (1):
        g(f(x)) -3|[==0,==1,>>0]-> p(p(h(f(x))))
        h(f(x)) -1|[==0,==1,>>0]-> p(h(f(x))) -1|0[==0,==1,>>0]-> p(p(h(f(x))))
       peak:
        h(f(x)) <-2|[==0,==1,==1]- f(x) -0|[==0,==1,==1]-> g(f(x))
       joins (1):
        h(f(x)) -1|[==0,==1,>>0]-> p(h(f(x))) -1|0[==0,==1,>>0]-> p(p(h(f(x))))
        g(f(x)) -3|[==0,==1,>>0]-> p(p(h(f(x))))
       Qed