Faithfulness of the spinning map for virtual links

Let $N$ be a fixed $n$-manifold, and let $\Phi_N:V_m\rightarrow V_{m+n}$ be the construction sending $(F\times [0,1],L)$ to $(F\times N\times [0,1],L\times N)$. Here $V_m$ denotes the collection of virtual $m$-links. Faithfulness conjecture. For any $N$, the map $\Phi_N:V_m\rightarrow V_{m+n}$ has the property that $\Phi_N(F\times [0,1],L)\cong \Phi_N(F'\times [0,1],L')$ only if $(F\times [0,1],L)$ and $(F'\times [0,1],L')$ are virtually equivalent up to mirror images and orientation reversal of $L$. This would assert that the spinning construction loses no information beyond these stated symmetries; whether this holds is left open in the source.