Approximate homoclinic relation from periodic historic measures

Let $f$ be a $C^r$ diffeomorphism of a compact manifold of dimension at least $2$, with $r\geq 1$. Let $\pi(P)$ and $\pi(Q)$ denote the minimal periods of two hyperbolic periodic points $P$ and $Q$.

Homoclinic-approximation conjecture. If there exist $x$ and $a,b\geq0$ with $a+b>1$ such that

$$\eta_x=a\cdot\frac{1}{\pi(P)}\sum_{j=1}^{\pi(P)}\delta_{f^jP}+b\cdot\frac{1}{\pi(P)}\sum_{j=1}^{\pi(Q)}\delta_{f^jQ},$$

then $f$ is accumulated in the $C^r$ topology by diffeomorphisms $g$ for which the continuations $P_g$ and $Q_g$ are homoclinically related.

This is proposed as a weakening of the earlier sufficient condition for heteroclinic connections. It remains open in the supplied material.