  Quantum spin liquids (QSL) are novel phases of matter which remain quantum
disordered even at the lowest temperature. They are characterized by emergent
gauge fields and fractionalized quasiparticles. Here we show that the
sub-Kelvin thermal transport of the three-dimensional $S=1/2$ hyper-hyperkagome
quantum magnet PbCuTe$_2$O$_6$ is governed by a sizeable charge-neutral
fermionic contribution which is compatible with the itinerant fractionalized
excitations of a spinon Fermi surface. We demonstrate that this hallmark
feature of the QSL state is remarkably robust against sample crystallinity,
large magnetic field, and field-induced magnetic order, ruling out the
imitation of QSL features by extrinsic effects. Our findings thus reveal the
characteristic low-energy features of PbCuTe$_2$O$_6$ which qualify this
compound as a true QSL material.
