  We investigate the $q$-holonomic properties of a class of link invariants
based on quantum group representations with vanishing quantum dimensions,
motivated by the search for the invariants' realization in physics. Some of the
best known invariants of this type, constructed from `typical' representations
of the unrolled quantum group $\mathcal U^H_{\zeta_{2r}}(\mathfrak{sl}_2)$ at a
$2r$-th root of unity, were introduced by Akutsu-Deguchi-Ohtsuki (ADO). We
prove that the ADO invariants for $r\geq 2$ are a $q$-holonomic family,
implying in particular that they satisfy recursion relations that are
independent of $r$. In the case of a knot, we prove that the $q$-holonomic
recursion ideal of the ADO invariants is contained in the recursion ideal of
the colored Jones polynomials, the subject of the celebrated AJ Conjecture.
(Combined with a recent result of S. Willetts, this establishes an isomorphism
of the ADO and Jones recursion ideals. Our results also confirm a recent
physically-motivated conjecture of Gukov-Hsin-Nakajima-Park-Pei-Sopenko.)
