Endpoint Fourier restriction and unrectifiability

We show that if a measure of dimension $s$ on ${\mathbb{R}}^d$ admits $(p,q)$ Fourier restriction for some endpoint exponents allowed by its dimension, namely $q=\frac{s}{d}{p}^{\mathrm{\prime }}$ for some $p>1$, then it is either absolutely continuous or $1$-purely unrectifiable.