  The study of CoNb$_2$O$_6$ sits at the confluence of simplicity and
complexity: on one hand, the model for Ising chains -- the building blocks of
CoNb$_2$O$_6$ -- in a transverse field, can be exactly solved and, thus, serves
as an archetype of quantum criticality; on the other hand, the weak but nonzero
interchain coupling adds geometric frustration to the stage, substantially
complicating the phase diagram. Here we utilize low-temperature specific heat
and thermal conductivity measurements to study the low-lying magnetic
excitations in CoNb$_2$O$_6$ and its spin-1 analogue NiNb$_2$O$_6$. The thermal
conductivity is found to be suppressed around the quantum critical point, where
the specific heat is enhanced due to gapless magnetic excitations, pointing to
the localized nature of the latter. These results highlight the predominant
role of frustration in determining the quantum critical magnetic excitations of
spin chains, which may furthermore underlie the remarkable similarities between
the phenomenology of these spin-1/2 and spin-1 systems.
