The nonexistence of a skein theory for the tangle group functor

Let $\mathcal{T}ang$ be the category of tangles, let $\Lambda\mathbf{Grp}$ be the category of groups, and let $\Gamma\colon\mathcal{T}ang\to\Lambda\mathbf{Grp}$ be the tangle group functor. A skein theory for $\Gamma$ is a finitely generated tensor ideal $\mathcal{I}$ of $\mathcal{T}ang$ such that $\Gamma$ factors through the canonical projection to $\mathcal{T}ang/\mathcal{I}$; write the induced functor as $\tilde{\Gamma}\colon\mathcal{T}ang/\mathcal{I}\to\Lambda\mathbf{Grp}$. Nonexistence conjecture. There is no skein theory for $\Gamma$: no finitely generated tensor ideal $\mathcal{I}$ of $\mathcal{T}ang$ has both the property that $\Gamma$ factors through $\mathcal{T}ang/\mathcal{I}$ and the property that $\tilde{\Gamma}$ is injective. The claim says that the tangle group functor cannot be localized by finitely many tensor-ideal relations while retaining injectivity on the resulting quotient. The supplied text gives no evidence of resolution, so the conjecture is recorded as open.