A note on higher integrability of projections

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