Weinberger's boundary-relative homotopy invariance conjecture for the rho-invariant

Let $F\colon X\to X'$ be a homotopy equivalence of manifolds with torsion-free fundamental groups that restricts to a diffeomorphism on the boundary. Equip the boundaries with Riemannian metrics $g$ and $g'$ such that the restriction of $F$ to the boundary is an isometry. Let $\alpha\colon \pi_1X\to U(n)$ be a unitary representation, with the corresponding representation on $X'$ understood via the homotopy equivalence. Weinberger's boundary-relative conjecture. For any such representation,

$$\rho(X,\alpha,g)=\rho(X',\alpha,g').$$

This extends Weinberger's homotopy-invariance conjecture to manifolds with boundary. The supplied text does not state whether this extension is known or refuted; the preceding discussion establishes related positive results for some groups and counterexamples to the unrestricted closed-manifold version.