Loops with invariant flexibility under the isostrophy

The question "Are the loops with universal (i.e. invariant under the isotopy of loops) flexibility law $xy\cdot x = x\cdot yx$, middle Bol loops?" is open in the theory of loops. If this conjecture is true then the loops for which all isostrophic loops are flexible are Moufang loops. In the present paper we prove that commutative loops with invariant flexibility under the isostrophy of loops are Moufang loops. In particular, we obtain that commutative $IP$-loops with universal flexibility are Moufang loops.