YES

Problem:
 strict:
  f(g(f(x))) -> f(g(g(g(f(x)))))
 weak:
  g(x) -> g(g(x))
  g(x) -> g(g(f(g(g(x)))))
  f(x) -> g(g(f(g(g(x)))))

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
             [1 0 0]  
   [g](x0) = [0 0 1]x0
             [0 0 0]  ,
   
             [1 2 0]     [0]
   [f](x0) = [2 2 2]x0 + [0]
             [0 0 0]     [1]
  orientation:
                [1 2 0]    [2]    [1 2 0]    [0]                   
   f(g(f(x))) = [2 4 0]x + [2] >= [2 4 0]x + [0] = f(g(g(g(f(x)))))
                [0 0 0]    [1]    [0 0 0]    [1]                   
   
          [1 0 0]     [1 0 0]           
   g(x) = [0 0 1]x >= [0 0 0]x = g(g(x))
          [0 0 0]     [0 0 0]           
   
          [1 0 0]     [1 0 0]                    
   g(x) = [0 0 1]x >= [0 0 0]x = g(g(f(g(g(x)))))
          [0 0 0]     [0 0 0]                    
   
          [1 2 0]    [0]    [1 0 0]                    
   f(x) = [2 2 2]x + [0] >= [0 0 0]x = g(g(f(g(g(x)))))
          [0 0 0]    [1]    [0 0 0]                    
  problem:
   strict:
    
   weak:
    g(x) -> g(g(x))
    g(x) -> g(g(f(g(g(x)))))
    f(x) -> g(g(f(g(g(x)))))
  Qed