The site-diluted compound (Yb$_{1-x}$Lu$_x$)$_4$As$_3$ is a scarce realization of the linear Heisenberg antiferromagnet partitioned into finite-size segments and is an ideal model compound for studying field-dependent effects of quenched disorder in the one-dimensional antiferromagnets. It differentiates from the systems studied so far in two aspects - the type of randomness and the nature of the energy gap in the pure sample. We have measured the specific heat of single-crystal (Yb$_{1-x}$Lu$_x$)$_4$As$_3$ in magnetic fields up to 19.5 T. The contribution $C_{perp}$ arising from the magnetic subsystem in an applied magnetic field perpendicular to the chains is determined. Compared to pure Yb$_4$As$_3$, for which $C_{perp}$ indicates a gap opening, for diluted systems a non-exponential decay is found at low temperatures which is consistent with the thermodynamic scaling of the specific heat established for a Bose-glass phase.
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