  In this note, we explore some consequences of the Modica-Mortola construction
of a singular elliptic measure, as regards the link between the quantitative
absolute continuity ($A_{\infty}$) of their approximations and the suitability
of a well-known tool, the so-called Kenig-Pipher condition
($\operatorname{KP}$). The Kenig-Pipher condition is used to ascertain absolute
continuity in the presence of some mild regularity of the coefficient matrix.
We perform some modifications of the Modica-Mortola example to show the
following two statements: (a) There are sequences of matrices for which both
$\operatorname{KP}$ and the $A_{\infty}$ condition break down in the limit. (b)
There are sequences of matrices for which $\operatorname{KP}$ breaks down but
$A_{\infty}$ is preserved in the limit.
