YES

Problem:
 p(f(f(x))) -> q(f(g(x)))
 p(g(g(x))) -> q(g(f(x)))
 q(f(f(x))) -> p(f(g(x)))
 q(g(g(x))) -> p(g(f(x)))

Proof:
 String Reversal Processor:
  f(f(p(x))) -> g(f(q(x)))
  g(g(p(x))) -> f(g(q(x)))
  f(f(q(x))) -> g(f(p(x)))
  g(g(q(x))) -> f(g(p(x)))
  Bounds Processor:
   bound: 1
   enrichment: match
   automaton:
    final states: {5}
    transitions:
     f1(15) -> 16*
     f1(11) -> 12*
     g1(12) -> 13*
     g1(14) -> 15*
     p1(21) -> 22*
     q1(10) -> 11*
     f0(5) -> 5*
     p0(5) -> 5*
     g0(5) -> 5*
     q0(5) -> 5*
     5 -> 21,10
     11 -> 14*
     13 -> 5*
     16 -> 5*
     22 -> 11*
   problem:
    
   Qed