Oriented-homotopy invariance conjecture for higher string topology operations

Let $M_1$ and $M_2$ be oriented manifolds and let $f:M_1\map M_2$ be an oriented homotopy equivalence, meaning a homotopy equivalence that sends the orientation of $M_1$ to an orientation of $M_2$. Consider the string operations in the paper's main theorem together with the conjectured relative string topology operations.

Oriented-homotopy invariance conjecture. The operations of the main theorem and the conjectured relative string topology operations are oriented-homotopy invariant.

This would extend the known oriented-homotopy invariance of the Chas–Sullivan BV structure to the higher and relative string topology operations constructed or proposed in the paper. The source gives no resolution of the conjecture.