Uniform projection dimension conjecture for planar Brownian level sets

Let $L^d$ be the Brownian level set and let $\mathcal{D}$ be the exceptional set defined in the source. For each direction $\theta\in[-\frac{\pi}{2},\frac{\pi}{2})$, write $\operatorname{proj}_{\theta}$ for projection in direction $\theta$, and let $\dim_H$ denote Hausdorff dimension. Uniform projection dimension conjecture. For every $\theta\in[-\frac{\pi}{2},\frac{\pi}{2})$,

$$\dim_H\bigl(\operatorname{proj}_{\theta}L^d\setminus\mathcal{D}\bigr)=2-\frac{d}{2}\quad\text{almost surely for }d=2,3.$$

The paper proves the asserted dimension in particular directions and conjectures that the same value holds uniformly for every direction; this is presented among the paper's further open problems.