In his seminal 1986 paper "A $q$-analogue of $U(\mathfrak{gl}(N+1))$, Hecke Algebra, and the Yang-Baxter Equation", Jimbo asserted (Proposition 3) that the quantum group associated to $\mathfrak{gl}_n$ and the Hecke algebra associated to $S_n$ are in Schur-Weyl duality. But the paper doesn't have the proofs, and Jimbo says in the introduction "Details will appear elsewhere".
Did the details ever actually appear? I haven't been able to find them anywhere, or even a citation to a longer paper.
I'm interested for curiosity's sake in Jimbo's original proof, but more broadly, I mostly just want a (hopefully clear) full proof of this result. Every source I've found either sketches the proof or does it for a related case.