Existence of knots distinguished by higher-order bordism invariants

Let $K$ be a knot, let $G$ be its group, let $M_K$ be the result of $0$--framed surgery on $K$, and let $P=\pi_1(M_K)$. For each $n\geq 0$, let $\mathcal{A}_i(K)$ denote the $i$th-order Alexander module of $K$, and let $\b_n(K)$ be its $n$th reduced bordism invariant, represented by

$$M_K\longrightarrow K(P/P^{(n+1)},1).$$

Existence conjecture. For each $n\geq 0$ there exist knots $K$ and $K_*$ such that $\mathcal{A}_i(K)\cong\mathcal{A}_i(K_*)$ for $0\leq i<n$, but

$$\b_n(K)\neq\b_n(K_*).$$

This would show that the higher-order bordism invariants contain information not detected by the lower-order Alexander modules. The supplied text does not state whether this existence claim has been proved or disproved.