Isomorphisms among quantum Grothendieck rings and propagation of positivity

Let ($mathfrak{g},mathsf{g})$ be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with $mathsf{g}$ being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories $mathscr{C}_{mathfrak{g}}$ and $mathscr{C}_{mathsf{g}}$ of finite-dimensional representations over the quantum loop algebras of $mathfrak{g}$ and $mathsf{g}$ respectively. As a consequence, we solve long-standing problems : the positivity of the analogs of Kazhdan-Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced $mathfrak{g}$. In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur-Weyl dualities, we prove the analog of Kazhdan-Lusztig conjecture (formulated in [H., Adv. Math., 2004]) for simple modules in remarkable monoidal subcategories of $mathscr{C}_{mathfrak{g}}$ for any non-simply-laced $mathfrak{g}$, and for any simple finite-dimensional modules in $mathscr{C}_{mathfrak{g}}$ for $mathfrak{g}$ of type $mathrm{B}_n$. In the course of the proof we obtain and combine several new ingredients. In particular we establish a quantum analog of $T$-systems, and also we generalize the isomorphisms of [H.-Leclerc, J. Reine Angew. Math., 2015] and [H.-O., Adv. Math., 2019] to all $mathfrak{g}$ in a unified way, that is isomorphisms between subalgebras of the quantum group of $mathsf{g}$ and subalgebras of the quantum Grothendieck ring of $mathscr{C}_mathfrak{g}$.