  Matrix-valued holomorphic quantum modular forms are intricate objects that
arise in successive refinements of the Volume Conjecture of knots and involve
three holomorphic, asymptotic and arithmetic objects. It is expected that the
algebraic properties of these objects can be deduced from the algebraic
properties of descendant state integrals, and we illustrate this for the case
of the $(-2,3,7)$-pretzel knot.
