Conjecture on BV stability for the general self-consistent system

Consider the self-consistent system in Theorem 3, with $F(x)>0$ for all $x\ne 0$ and

$$F'(x)=O\left(\frac{1}{|\log x|}\right)\quad\text{as }x\to 0+.$$

Let the uniform density denote the density of Lebesgue measure, and let an acim mean an absolutely continuous invariant measure with its invariant density. For a density $f$, write

$$\|f\|_{BV}=\operatorname{var}(f)+\int |f|.$$

BV stability conjecture. The uniform density is stable with respect to the $BV$-norm for all $\varepsilon>0$, while every other acim is unstable with respect to the $BV$-norm.

This concerns the stability of the invariant densities in the regime where Theorem 3 guarantees uniqueness for sufficiently small self-consistency and multiple acims for sufficiently large self-consistency. The stability assertions are based on the authors' computer simulations and remain conjectural.