Continuity conjecture for projection eigenfunctions in fractal percolations

Let $F_\alpha$ be the operator defined in the paper, let $\mathcal{I}$ be the index set of the construction, let $p_{\mathbf{i}}$ be the corresponding probabilities, and let $M$ denote the normalization parameter. For each direction $\alpha\in[0,\pi]$, consider the eigenfunction $f_\alpha$ of $F_\alpha$. Continuity conjecture. The operator $F_\alpha$ has an eigenfunction $f_\alpha$ with eigenvalue

$$\frac{\sum_{\mathbf{i}\in\mathcal{I}}p_{\mathbf{i}}}{M},$$

which is continuous for all $\alpha\in[0,\pi]$, except for a set of directions of Hausdorff dimension $0$. This conjecture concerns the regularity of projection densities in fractal percolation; the preceding discussion establishes continuity and interval projections in substantial ranges of directions, while exceptional directions remain the main issue.