Infinite Bernoulli convolutions with different probabilities

$Y_{\lambda}^p$:=$\sum_{n=0}^{\infty}\pm \lambda ^n$

where the "$+$" and "$-$" signs are chosen independently with probability $p$ and $1-p$. Let $\nu_\lambda^p$ be the distribution of the random sum $\nu_\lambda^p(E)$:=$Prob(Y_{\lambda}^p \in E)$ for every set $E$. The conjecture is that for every $p \in (0,1)$ the measure $\nu_\lambda^p$ is absolutely continuous w.r.t. Lebesgue measure and with the density in $L^2(R)$ for almost every $\lambda\in (p^p\cdot(1-p)^{(1-p)},1).$
    B. Solomyak and Y. Peres [3, [Corollary 1.4] proved that for every $p \in (\frac{1}{3},\frac{2}{3})$ the distribution $\nu_\lambda^p$ is absolutely continuous with $L^2(R)$ density for almost every $\lambda \in (p^2+(1-p)^2,1).$ In this paper we extend the parameter interval where a weakened version of the conjecture still holds. Namely, we prove Corollary 3 that for every $p \in (0,\frac{1}{3}]$ the measure $\nu_\lambda^p$ is absolutely continuous with $L^2(R)$ density for almost every $\lambda\in(F(p),1)$, where $F(p)=(1-2p)^{2-\log41/\log 9}$, see Figure 3.