  We treat the $\tau$-tilting finiteness of those minimal
representation-infinite (min-rep-infinite) algebras which are non-distributive.
Building upon the new results of Bongartz, we fully determine which algebras in
this family are $\tau$-tilting finite and which ones are not. This complements
our previous work in which we carried out a similar analysis for the
min-rep-infinite biserial algebras. Consequently, we obtain nontrivial explicit
sufficient conditions for $\tau$-tilting infiniteness of a large family of
algebras. This also produces concrete families of "minimal $\tau$-tilting
infinite algebras"-- the modern counterpart of min-rep-infinite algebras,
independently introduced by the author and Wang.
  We further use our results on the family of non-distributive algebras to
establish a conjectural connection between the $\tau$-tilting theory and two
geometric notions in the study of module varieties introduced by Chindris,
Kinser and Weyman. We verify the conjectures for the algebras studied in this
note: For the min-rep-infinite algebras which are non-distributive or biserial,
we show that if $\Lambda$ has the dense orbit property, then it must be
$\tau$-tilting finite. Moreover, we prove that such an algebra is
Schur-representation-finite if and only if it is $\tau$-tilting finite. The
latter result gives a categorical interpretation of
Schur-representation-finiteness over this family of min-rep-infinite algebras.
