A note on higher integrability of projections

Let $t\in [1,2)$ and $p>2\mathrm{/}(2-t)$. I construct a $t$-Frostman Borel measure $\mu$ on $[0,1{]}^2$ such that ${\pi }_{\theta }\mu \mathrm{\notin }{L}^p$ for every $\theta \in S^1$. This answers a question of Peres and Schlag.