Let $a\in (0,4]$ be the logistic family $Q_a:[0,1] \to [0,1]$, $Q_a(x) = ax(1-x).$ In the book "One-Dimensional Dynamics - W. de Melo, S. van Strien", it is stated that the set $$\mathcal C =\{a\in (0,4]; \mbox{ $Q_a$ admits an invariant measure $\mu_a$, such that $\mu_a(\mathrm dx) \ll \mathrm{Lebesgue}(\mathrm dx)$}\},$$ has $4$ as an accumulation point (see the result at the end of this question).
I searched online if there is any information about the support of the measures $\{\mu_a\}_{a\in \mathcal C}.$ I believe that there exists a subsequence $\{a_n\}_{n\in\mathbb N} \subset \mathcal C,$ such that $\lim_{n\to \infty} a_n= 4$ and supp $(\mu_{a_n}) \to [0,1]$ (in the Hausforff metric), however I could neither find a proof from this fact nor prove it by myself.
Does anyone have any suggestions for attacking this problem or a reference?
Corollary (Jakobson). Let $Q_a : [0,1] \to [0,1]$, $a \in (0,4]$, be the quadratic family $Q_a(x) = ax(1-x)$. There exists a subset $\mathcal{C} \subset (0,4]$ of positive Lebesgue measure with the following properties:
If $a \in \mathcal{C}$ then $Q_a$ has an absolutely continuous invariant probability measure with positive entropy.
The parameter value $a=4$ is a Lebesgue density point of $\mathcal{C}$, namely,
$$\lim_{\epsilon \to 0} \frac{\lambda(\mathcal{C} \cap [4-\epsilon,4])}{\epsilon} = 1.$$
The paper Absolutely Continuous Invariant Measures for One-Parameter Families of One-Dimensional Maps - M. V. Jakobson gives a positive answer for the existence of the sequence $\{a_n\}_{n\in \mathbb N}$ conjecture on the question (see Remark XIII/5 on page 87).