The change in the configuration of valence protons between the initial and final states in the neutrinoless double-$beta$ decay of $^{130}$Te $rightarrow$ $^{130}$Xe and of $^{136}$Xe $rightarrow$ $^{136}$Ba has been determined by measuring the cross sections of the ($d$,$^3$He) reaction with 101-MeV deuterons. Together with our recent determination of the relevant neutron configurations involved in the process, a quantitative comparison with the latest shell-model and interacting-boson-model calculations reveals significant discrepancies. These are the same calculations used to determine the nuclear matrix elements governing the rate of neutrinoless double-$beta$ decay in these systems.
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