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=2
  
  interpretation:
                  [0]
   [g](x0) = x0 + [1],
   
                  [0]
   [f](x0) = x0 + [1],
   
                     [2 1]     [2 1]     [2 1]     [0]
   [t](x0, x1, x2) = [0 0]x0 + [0 0]x1 + [0 0]x2 + [1]
  orientation:
                       [2 1]    [2 1]    [2 1]    [3]    [2 1]    [2 1]    [2 1]    [2]                 
   t(f(x),g(y),f(z)) = [0 0]x + [0 0]y + [0 0]z + [1] >= [0 0]x + [0 0]y + [0 0]z + [1] = t(z,g(x),g(y))
   
                       [2 1]    [2 1]    [2 1]    [3]    [2 1]    [2 1]    [2 1]    [2]                 
   t(g(x),g(y),f(z)) = [0 0]x + [0 0]y + [0 0]z + [1] >= [0 0]x + [0 0]y + [0 0]z + [1] = t(f(y),f(z),x)
   
                 [0]        [0]          
   f(g(x)) = x + [2] >= x + [2] = g(f(x))
   
                 [0]        [0]          
   g(f(x)) = x + [2] >= x + [2] = f(g(x))
   
                 [0]        [0]          
   f(f(x)) = x + [2] >= x + [2] = g(g(x))
   
                 [0]        [0]          
   g(g(x)) = x + [2] >= x + [2] = f(f(x))
  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