Keleti's line segment extension conjecture

Let $L$ be a set of line segments in $\mathbb{R}^n$. For each line segment $\ell$, let $\widetilde\ell$ denote the line containing $\ell$. Keleti's line segment extension conjecture.

$$\mathrm{dim}\,\left(\bigcup_{\ell\in L}\widetilde\ell\right)=\mathrm{dim}\,\left(\bigcup_{\ell\in L}\ell\right).$$

This conjecture relates the dimensions of unions of line segments and their containing lines and is presented as closely related to the Tube Doubling Conjecture. The supplied material does not state its resolution status.