Absolute continuity in families of parametrised non-homogeneous self-similar measures

In 2016, Shmerkin and Solomyak showed that if $U \subset \mathbb{R}$ is an interval, and $\{\mu_{u}\}_{u \in U}$ is an analytic family of homogeneous self-similar measures on $\mathbb{R}$ with similitude dimensions exceeding one, then, under a mild transversality assumption, $\mu_{u} \ll \mathcal{L}^{1}$ for all parameters $u \in U \setminus E$, where $\dim_{\mathrm{H}} E = 0$. The purpose of this paper is to generalise the result of Shmerkin and Solomyak to non-homogeneous self-similar measures. As a corollary, we obtain new information about the absolute continuity of projections of non-homogeneous planar self-similar measures.