  Relying on results due to Shmerkin and Solomyak, we show that outside a
$0$-dimensional set of parameters, for every planar homogeneous self-similar
measure $\nu$, with strong separation, dense rotations and dimension greater
than $1$, there exists $q>1$ such that $\{P_{z}\nu\}_{z\in S}\subset
L^{q}(\mathbb{R})$. Here $S$ is the unit circle and $P_{z}w=\left\langle
z,w\right\rangle $ for $w\in\mathbb{R}^{2}$. We then study such measures. For
instance, we show that $\nu$ is dimension conserving in each direction and that
the map $z\rightarrow P_{z}\nu$ is continuous with respect to the weak topology
of $L^{q}(\mathbb{R})$.
