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In this note, we explore some consequences of the Modica-Mortola constructionof a singular elliptic measure, as regards the link between the quantitativeabsolute continuity ($A_{\infty}$) of their approximations and the suitabilityof a well-known tool, the so-called Kenig-Pipher condition($\operatorname{KP}$). The Kenig-Pipher condition is used to ascertain absolutecontinuity in the presence of some mild regularity of the coefficient matrix.We perform some modifications of the Modica-Mortola example to show thefollowing 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.

Failure to slide: a brief note on the interplay between the Kenig-Pipher condition and the absolute continuity of elliptic measures