Dirichlet is not just bad and singular in many rational IFS fractals

For $m\ge 2$, consider $K$ the $m$-fold Cartesian product of the limit set of an IFS of two affine maps with rational coefficients. If the contraction rates of the IFS are reciprocals of integers, and $K$ does not degenerate to singleton, we construct vectors in $K$ that lie within the ``folklore set'' as defined by Beresnevich et al., meaning they are Dirichlet improvable but not singular or badly approximable (in fact our examples are Liouville vectors). We further address the topic of lower bounds for the Hausdorff and packing dimension of these folklore sets within $K$, however we do not compute bounds explicitly. Our class of fractals extends (Cartesian products of) classical missing digit fractals, for which analogous results had recently been obtained.