  Arguably the most favorable situation for spins to enter the long-sought
quantum spin liquid (QSL) state is when they sit on a kagome lattice. No
consensus has been reached in theory regarding the true ground state of this
promising platform. The experimental efforts, relying mostly on one archetypal
material ZnCu$_3$(OH)$_6$Cl$_2$, have also led to diverse possibilities. Apart
from subtle interactions in the Hamiltonian, there is the additional degree of
complexity associated with disorder in the real material ZnCu$_3$(OH)$_6$Cl$_2$
that haunts most experimental probes. Here we resort to heat transport
measurement, a cleaner probe in which instead of contributing directly, the
disorder only impacts the signal from the kagome spins. For
ZnCu$_3$(OH)$_6$Cl$_2$ and a related QSL candidate Cu$_3$Zn(OH)$_6$FBr, we
observed no contribution by any spin excitation nor any field-induced change to
the thermal conductivity. These results impose different constraints on various
scenarios about the ground state of these two kagome compounds: while a gapped
QSL, or certain quantum paramagnetic state other than a QSL, is compatible with
our results, a gapless QSL must be dramatically modified by the disorder so
that gapless spin excitations are localized.
