Meagreness conjecture for Dirac physical measures on saddle-type hyperbolic periodic points

Let $M$ be the compact manifold underlying the space of diffeomorphisms, and let $\operatorname{Diff}^1(M)$ denote the space of $C^1$ diffeomorphisms of $M$. A saddle-type hyperbolic periodic point is a hyperbolic periodic point that is neither a sink nor a source. A Dirac physical measure is a physical measure supported on a single fixed or periodic point. Meagreness conjecture. The set of $f\in\operatorname{Diff}^1(M)$ having a Dirac physical measure supported on a saddle-type hyperbolic fixed or periodic point is meagre, that is, a countable union of closed sets with empty interior. This is proposed as a testing conjecture for the question of which dynamical mechanisms produce such measures and how they relate to homoclinic tangencies. The examples discussed in the source suggest that this phenomenon is highly non-generic, but the conjecture is presented without a resolution.