The visible part of plane self-similar sets

Given a compact subset $F$ of ${\mathbb{R}}^2$, the visible part $V_{\theta }F$ of $F$ from direction $\theta$ is the set of $x$ in $F$ such that the half-line from $x$ in direction $\theta$ intersects $F$ only at $x$. It is suggested that if ${\dim }_{H}F\ge 1$ then ${\dim }_H{V}_{\theta }F=1$ for almost all $\theta$, where ${\dim }_H$ denotes Hausdorff dimension. We confirm this when $F$ is a self-similar set satisfying the convex open set condition and such that the orthogonal projection of $F$ onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and $F$ need not be connected.