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:
   [g](x0, x1) = x0 + 2x1,
   
   [f](x0, x1) = 2x0 + x1 + 1,
   
   [h](x0, x1) = x0 + 4x1 + 4
  orientation:
   g(f(x,y),z) = 2x + y + 2z + 1 >= 2x + y + 2z + 1 = f(x,g(y,z))
   
   g(h(x,y),z) = x + 4y + 2z + 4 >= x + 4y + 2z + 2 = g(x,f(y,z))
   
   g(x,h(y,z)) = x + 2y + 8z + 8 >= x + 2y + 4z + 4 = h(g(x,y),z)
  problem:
   g(f(x,y),z) -> f(x,g(y,z))
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                  [1 1 0]     [1 0 0]  
    [g](x0, x1) = [0 1 1]x0 + [0 0 0]x1
                  [0 0 0]     [1 0 0]  ,
    
                  [1 0 0]          [0]
    [f](x0, x1) = [0 0 0]x0 + x1 + [1]
                  [0 0 0]          [0]
   orientation:
                  [1 0 0]    [1 1 0]    [1 0 0]    [1]    [1 0 0]    [1 1 0]    [1 0 0]    [0]              
    g(f(x,y),z) = [0 0 0]x + [0 1 1]y + [0 0 0]z + [1] >= [0 0 0]x + [0 1 1]y + [0 0 0]z + [1] = f(x,g(y,z))
                  [0 0 0]    [0 0 0]    [1 0 0]    [0]    [0 0 0]    [0 0 0]    [1 0 0]    [0]              
   problem:
    
   Qed