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We report low-temperature specific heat, $C(T)$, measurements on (Yb$_{1-x}$Lu$_x$)$_4$As$_3$ with $x=0.01$ and $x=0.03$, where nonmagnetic Lu atoms are randomly distributed on antiferromagnetic $S=1/2$ Heisenberg chains with $J/k_{\mathrm B}=28$ K. The observed reduction of $C$ below 15 K with increasing $x$ is accurately described by quantum transfer matrix simulations without any adjustable parameter, implying that the system is an excellent experimental realization of segmented quantum spin chains. Finite-size effects consistent with conformal-field theory predictions are leading to the formation of an effective low-energy gap. The size of the gap increases with Lu content and accounts for the impurity driven reduction of the specific heat. For both concentrations our results verify experimentally the low temperature scaling behavior established theoretically and also confirm the value of $J$ determined from pure Yb$_4$As$_3$., Comment: 7 pages, 6 figures