The exact dimension formula for line sets of trapped sets

Let $X \subset \mathbb{R}^n$ and let $T(X)=T$ be the trapping number of $X$. Suppose that $H \in \mathcal{A}(n,T)$ is the unique hyperplane such that

$$\dim(X \setminus H)<\dim X,$$

and set $X_1=X \cap H$ and $X_2=X \setminus H$. Here, $\mathcal{L}(X)$ denotes the set of lines intersecting $X$, and $\dim$ denotes Hausdorff dimension.

Exact dimension formula. One has

$$\dim \mathcal{L}(X)=\max\left\{\dim(X_1 \times X_2),\min\left\{\dim(X_1 \times X_1),2(T-1)\right\}\right\}.$$

The formula is proposed because the preceding lower bound is attained by an example, but the authors do not expect that bound to be sharp for all Borel sets. The conjecture gives an exact expression in the special situation where the trapping hyperplane is unique.