YES

Problem:
 strict:
  t(f(x),g(y),f(z())) -> t(z(),g(x),g(y))
  t(g(x),g(y),f(z())) -> t(f(y),f(z()),x)
 weak:
  f(g(x)) -> g(f(x))
  g(f(x)) -> f(g(x))
  f(f(x)) -> g(g(x))
  g(g(x)) -> f(f(x))

Proof:
 Matrix Interpretation Processor: dim=5
  
  interpretation:
             [1 1 1 1 0]  
             [0 0 0 1 0]  
   [g](x0) = [0 0 0 0 1]x0
             [0 0 0 0 0]  
             [0 0 0 0 0]  ,
   
                     [1 0 1 1 0]     [1 0 1 0 0]     [1 0 1 0 1]     [1]
                     [1 0 1 0 1]     [1 1 1 0 0]     [0 0 1 0 0]     [1]
   [t](x0, x1, x2) = [0 1 0 0 0]x0 + [0 0 0 0 0]x1 + [0 0 0 0 0]x2 + [0]
                     [0 1 0 0 0]     [0 0 0 0 0]     [0 0 0 0 0]     [0]
                     [0 1 0 0 0]     [0 0 0 0 0]     [0 0 0 0 0]     [0],
   
             [1 0 1 0 1]  
             [0 0 0 0 0]  
   [f](x0) = [0 1 0 1 0]x0
             [0 0 0 0 0]  
             [0 0 0 1 0]  ,
   
         [0]
         [0]
   [z] = [0]
         [1]
         [0]
  orientation:
                         [1 1 1 1 1]    [1 1 1 1 1]    [3]    [1 1 1 1 1]    [1 1 1 1 1]    [2]                   
                         [1 1 1 2 1]    [1 1 1 2 1]    [2]    [1 1 1 2 1]    [0 0 0 0 1]    [1]                   
   t(f(x),g(y),f(z())) = [0 0 0 0 0]x + [0 0 0 0 0]y + [0] >= [0 0 0 0 0]x + [0 0 0 0 0]y + [0] = t(z(),g(x),g(y))
                         [0 0 0 0 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]                   
                         [0 0 0 0 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]                   
   
                         [1 1 1 1 1]    [1 1 1 1 1]    [3]    [1 0 1 0 1]    [1 1 1 1 1]    [2]                   
                         [1 1 1 1 1]    [1 1 1 2 1]    [2]    [0 0 1 0 0]    [1 1 1 2 1]    [2]                   
   t(g(x),g(y),f(z())) = [0 0 0 1 0]x + [0 0 0 0 0]y + [0] >= [0 0 0 0 0]x + [0 0 0 0 0]y + [0] = t(f(y),f(z()),x)
                         [0 0 0 1 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]                   
                         [0 0 0 1 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]                   
   
             [1 1 1 1 1]     [1 1 1 1 1]           
             [0 0 0 0 0]     [0 0 0 0 0]           
   f(g(x)) = [0 0 0 1 0]x >= [0 0 0 1 0]x = g(f(x))
             [0 0 0 0 0]     [0 0 0 0 0]           
             [0 0 0 0 0]     [0 0 0 0 0]           
   
             [1 1 1 1 1]     [1 1 1 1 1]           
             [0 0 0 0 0]     [0 0 0 0 0]           
   g(f(x)) = [0 0 0 1 0]x >= [0 0 0 1 0]x = f(g(x))
             [0 0 0 0 0]     [0 0 0 0 0]           
             [0 0 0 0 0]     [0 0 0 0 0]           
   
             [1 1 1 2 1]     [1 1 1 2 1]           
             [0 0 0 0 0]     [0 0 0 0 0]           
   f(f(x)) = [0 0 0 0 0]x >= [0 0 0 0 0]x = g(g(x))
             [0 0 0 0 0]     [0 0 0 0 0]           
             [0 0 0 0 0]     [0 0 0 0 0]           
   
             [1 1 1 2 1]     [1 1 1 2 1]           
             [0 0 0 0 0]     [0 0 0 0 0]           
   g(g(x)) = [0 0 0 0 0]x >= [0 0 0 0 0]x = f(f(x))
             [0 0 0 0 0]     [0 0 0 0 0]           
             [0 0 0 0 0]     [0 0 0 0 0]           
  problem:
   strict:
    
   weak:
    f(g(x)) -> g(f(x))
    g(f(x)) -> f(g(x))
    f(f(x)) -> g(g(x))
    g(g(x)) -> f(f(x))
  Qed