YES

Problem:
 g(f(x,y),z) -> f(x,g(y,z))
 g(h(x,y),z) -> g(x,f(y,z))
 g(x,h(y,z)) -> h(g(x,y),z)

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [h](x0, x1) = x0 + 7x1 + 4,
   
   [g](x0, x1) = x0 + 4x1,
   
   [f](x0, x1) = x0 + x1 + 1
  orientation:
   g(f(x,y),z) = x + y + 4z + 1 >= x + y + 4z + 1 = f(x,g(y,z))
   
   g(h(x,y),z) = x + 7y + 4z + 4 >= x + 4y + 4z + 4 = g(x,f(y,z))
   
   g(x,h(y,z)) = x + 4y + 28z + 16 >= x + 4y + 7z + 4 = h(g(x,y),z)
  problem:
   g(f(x,y),z) -> f(x,g(y,z))
   g(h(x,y),z) -> g(x,f(y,z))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [h](x0, x1) = x0 + 4x1 + 1,
    
    [g](x0, x1) = x0 + x1 + 6,
    
    [f](x0, x1) = 4x0 + x1
   orientation:
    g(f(x,y),z) = 4x + y + z + 6 >= 4x + y + z + 6 = f(x,g(y,z))
    
    g(h(x,y),z) = x + 4y + z + 7 >= x + 4y + z + 6 = g(x,f(y,z))
   problem:
    g(f(x,y),z) -> f(x,g(y,z))
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                   [1 1 0]     [1 1 0]  
     [g](x0, x1) = [0 0 1]x0 + [0 0 0]x1
                   [0 1 0]     [0 0 0]  ,
     
                   [1 0 0]     [1 1 0]     [0]
     [f](x0, x1) = [0 1 0]x0 + [0 0 1]x1 + [1]
                   [0 1 0]     [0 1 0]     [1]
    orientation:
                   [1 1 0]    [1 1 1]    [1 1 0]    [1]    [1 0 0]    [1 1 1]    [1 1 0]    [0]              
     g(f(x,y),z) = [0 1 0]x + [0 1 0]y + [0 0 0]z + [1] >= [0 1 0]x + [0 1 0]y + [0 0 0]z + [1] = f(x,g(y,z))
                   [0 1 0]    [0 0 1]    [0 0 0]    [1]    [0 1 0]    [0 0 1]    [0 0 0]    [1]              
    problem:
     
    Qed