Let $K, K'$ be knots in $S^3$, and $T, T'$ the boundaries of their tubular neighborhoods.

Recall that by theorems of Waldhausen, and Gordon and Luecke, one knows the following: an isomorphism $[\pi_1(T) \to \pi_1(S^3 \setminus K)] \cong [(\pi_1(T') \to \pi_1(S^3 \setminus K')]$ implies that $K$ and $K'$ are isotopic.

Suppose the category of linear representations of $[\pi_1(T) \to \pi_1(S^3 \setminus K)]$ is equivalent to the category of linear representations of $[\pi_1(T') \to \pi_1(S^3 \setminus K')]$. Are $K$ and $K'$ isotopic?

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Note: In a previous version of this question, I asked whether this is possible if you interpret the category above as a tensor category; in the comments it is pointed out that then, yes, at least if you allow big representations. But, now I realize that in fact I don't have the tensor product structure.