  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 $\mathscr{C}_\mathfrak{g}$.
