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The recently observed $\alpha$-decay chain${}^{108}\text{Xe}\to{}^{104}\text{Te}\to{}^{100}\text{Sn}$ [K.~Auranen\emph{et al.}, Phys.\ Rev.\ Lett.\ {\bf121}, 182501 (2018)] could providevaluable information on the $\alpha$ clustering in even-even nuclei slightlyabove the major shells $Z=50$ and $N=50$. In this work, this $\alpha$-decaychain is studied theoretically within the framework of the density-dependentcluster model plus the two-potential approach. We calculate the $\alpha$-decayhalf-lives of ${}^{104}$Te and ${}^{108}$Xe and study in detail theirdependence on the $Q$ value and the density-profile parameters of the corenucleus. Various physical properties of ${}^{104}$Te, the heaviest nucleus witha doubly magic self-conjugate core $+$ $\alpha$, are calculated, with twodifferent assumptions on the renormalization factor of the double-foldingpotential, which could be a useful reference for future experimental studies.

$\alpha$ Clustering slightly above ${}^{100}$Sn in the Light of the New Experimental Data on the Superallowed $\alpha$ Decay