On the distribution of indicators of unconditional exponential bases in spaces with a power weight

In the present paper we consider the existence of unconditional exponential bases in a space of locally integrable functions on a bounded interval of the real number line $I$ satisfying
$$ \|f\|:=\sqrt{\int_I|f(t)|^2e^{-2h(t)}\,dt}<\infty, $$
where $h(t)$ is a convex function on this interval. The lower estimate was obtained for the frequency of indicators of unconditional bases of exponentials when $I=(-1;1)$, $h(t)=-\alpha\ln(1-|t|)$, $\alpha>0$.