The $SL_2$ Kakeya conjecture

Let $\mathcal L_{SL_2}$ be the set of lines in $\varmathbb{R}^3$ of the form $\ell_{(a,b,c,d)}=(a,b,0)+\operatorname{span}(c,d,1)$ with $ad-bc=1$, and let $\mathcal L_{SL_2}^*$ also include lines of the form $(0,0,t)+\operatorname{span}(c,d,0)$. An $SL_2$ Kakeya set is a compact set $K\subset\varmathbb{R}^3$ containing a unit line segment in every direction except $(0,0,1)$, with each segment lying on a line in $\mathcal L_{SL_2}^*$. The $SL_2$ Kakeya conjecture. Every $SL_2$ Kakeya set in $\varmathbb{R}^3$ has Minkowski and Hausdorff dimension $3$. This is a special case of the three-dimensional Kakeya set conjecture. It was recently resolved by Fässler and Orponen, and the paper proves the result independently by different methods.