The Fourier restriction conjecture for Frostman measures on the parabola

Let $\mathbb{P}$ denote the parabola, and let $B(R)$ be the ball of radius $R$ in frequency space. For $0\leq s\leq 1$, let $\mu$ be a Borel measure on $\mathbb{P}$ satisfying the $s$-Frostman condition

$$\mu(B(x,r))\leq r^s\qquad\text{for all }x\in\mathbb{R}^2\text{ and }r>0.$$

Fourier restriction conjecture for Frostman measures on the parabola. For every $0\leq s\leq 1$ and $\epsilon>0$, there exists $p=p(\epsilon,s)\geq 1$ such that

$$\|\widehat{\mu}\|_{L^p(B(R))}\leq C_{\epsilon,s}R^{[2-\min\{3s,1+s\}]/p+\epsilon},\qquad R\geq 1.$$

This conjecture asks for the optimal power of $R$, up to an arbitrarily small loss, in Fourier $L^p$ estimates for measures supported on the parabola. The source states that the precise value of $\epsilon$ is not known for every pair $(p,s)$ with $s\in[0,1]$ and $p>4$, so the proposed assertion remains open.