YES

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

Proof:
 Church Rosser Transformation Processor (kb):
  f(g(x)) -> h(g(x),g(x))
  f(s(x)) -> h(s(x),s(x))
  g(x) -> s(x)
  critical peaks: joinable
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [f](x0) = 5x0 + 4,
     
     [s](x0) = x0,
     
     [g](x0) = x0 + 1,
     
     [h](x0, x1) = x0 + 4x1 + 4
    orientation:
     f(g(x)) = 5x + 9 >= 5x + 9 = h(g(x),g(x))
     
     f(s(x)) = 5x + 4 >= 5x + 4 = h(s(x),s(x))
     
     g(x) = x + 1 >= x = s(x)
    problem:
     f(g(x)) -> h(g(x),g(x))
     f(s(x)) -> h(s(x),s(x))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [f](x0) = 6x0,
      
      [s](x0) = 5x0,
      
      [g](x0) = x0 + 4,
      
      [h](x0, x1) = 2x0 + 2x1
     orientation:
      f(g(x)) = 6x + 24 >= 4x + 16 = h(g(x),g(x))
      
      f(s(x)) = 30x >= 20x = h(s(x),s(x))
     problem:
      f(s(x)) -> h(s(x),s(x))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [f](x0) = 4x0 + 7,
       
       [s](x0) = 3x0,
       
       [h](x0, x1) = x0 + 3x1 + 6
      orientation:
       f(s(x)) = 12x + 7 >= 12x + 6 = h(s(x),s(x))
      problem:
       
      Qed