Three-halves logarithmic dimension conjecture for Furstenberg sets

Let $\mathfrak{h}$ and $\mathfrak{g}$ be dimension functions with

$$\mathfrak{h}(x)=\mathfrak{g}(x)=\frac{1}{\log(1/x)},$$

and let $E\subseteq\mathbb{R}^{2}$ be an $F_{\mathfrak{h}\mathfrak{g}}$-set: there is a subset $L$ of the unit circle with $\mathcal{H}^{\mathfrak{g}}(L)>0$, and for every direction $e\in L$ there is a line segment $\ell_e$ in direction $e$ such that $\mathcal{H}_{\delta}^{\mathfrak{h}}(\ell_e\cap E)>1$ for all sufficiently small $\delta$.

Three-halves logarithmic dimension conjecture. The function

$$\frac{1}{\log^{3/2}(1/x)}$$

should be an appropriate dimension function for $E$, in the sense that a logarithmic gap can be estimated.

The conjecture asks for the expected logarithmic refinement of the preceding dimension estimates in the case where both direction and fibre dimension functions equal $1/\log(1/x)$. The source does not state whether this prediction has been proved or disproved.