Here is the input problem:
  Axiom 1 (diag): mul(X, X) = mul(Y, Y).
  Axiom 2 (flattening): mul3 = mul(y, x2).
  Axiom 3 (flattening): mul2 = mul(y, y).
  Axiom 4 (flattening): mul4 = mul(mul(y, y), mul(y, x2)).
  Axiom 5 (eq1323): X = mul(Y, mul(mul(mul(Y, Y), X), Y)).
  Axiom 6 (flattening): mul5 = mul(mul(mul(y, y), mul(y, x2)), y).
  Goal 1 (eq2744): x = mul(mul(mul(y, y), mul(y, x2)), y).

1. mul(X, X) ~> mul(?, ?)
2. mul(y, x2) -> mul3
3. mul(?, ?) -> mul2
4. mul(mul2, mul3) -> mul4
5. mul(X, mul(mul(mul2, Y), X)) -> Y
6. mul(mul4, y) -> mul5
7. mul(X, mul(mul2, X)) -> mul2
8. mul(mul3, mul4) -> mul2
9. mul(X, mul(mul4, X)) -> mul3
10. mul3 -> mul4
11. mul(mul4, mul2) -> mul4
12. mul(y, mul5) -> mul4
13. mul(mul2, mul4) -> mul4
14. mul(mul(mul2, X), mul2) -> X
15. mul(X, mul2) -> X
16. mul(mul2, X) -> X
17. mul(X, mul(Y, X)) -> Y
18. mul(x2, mul4) -> y
19. x2 -> mul5
20. mul(mul5, mul4) -> y
21. mul(mul(X, Y), X) -> Y

Ran out of critical pairs. This means the conjecture is not true.
Here is the final rewrite system:
  x2 -> mul5
  mul3 -> mul4
  mul(X, X) ~> mul(?, ?)
  mul(X, mul2) -> X
  mul(?, ?) -> mul2
  mul(mul5, mul4) -> y
  mul(y, mul5) -> mul4
  mul(mul4, y) -> mul5
  mul(mul2, X) -> X
  mul(X, mul(Y, X)) -> Y
  mul(mul(X, Y), X) -> Y

RESULT: CounterSatisfiable (the conjecture is false).