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This work is motivated by an article by Wang, Casati, and Prosen[Phys. Rev. E vol. 89, 042918 (2014)] devoted to a study of ergodicityin two-dimensional irrational right-triangular billiards. Numericalresults presented there suggest that these billiards are generally notergodic. However, they become ergodic when the billiard angle is equalto \pi/2 π / 2 times a Liouvillian irrational, morally a class of irrational numberswhich are well approximated by rationals. In particular, Wang etal. study a special integer counter that reflects the irrationalcontribution to the velocity orientation; they conjecture that thiscounter is localized in the generic case, but grows in the Liouvilliancase. We propose a generalization of the Wang-Casati-Prosen counter:this generalization allows to include rational billiards intoconsideration. We show that in the case of a 45°\!\!:\!45°\!\!:\!90° 45 ° : 45 ° : 90 ° billiard, the counter grows indefinitely, consistent with theLiouvillian scenario suggested by Wang et al.