We report measurements of in-plane electrical and thermal transport properties in the limit $T rightarrow 0$ near the unconventional quantum critical point in the heavy-fermion metal $beta$-YbAlB$_4$. The high Kondo temperature $T_K$ $simeq$ 200 K in
this material allows us to probe transport extremely close to the critical point, at unusually small values of $T/T_K < 5 times 10^{-4}$. Here we find that the Wiedemann-Franz law is obeyed at the lowest temperatures, implying that the Landau quasiparticles remain intact in the critical region. At finite temperatures we observe a non-Fermi liquid T-linear dependence of inelastic scattering processes to energies lower than those previously accessed. These processes have a weaker temperature dependence than in comparable heavy fermion quantum critical systems, and suggest a new temperature scale of $T sim 0.3 K$ which signals a sudden change in character of the inelastic scattering.
