Conjecture on the sharp three-dimensional multijoints constant

For a multijoints configuration $(\mathcal J,\mathcal L_1,\mathcal L_2,\mathcal L_3)$ in $\mathbb F^3$, let $|\mathcal L_i|$ denote the size of the $i$th line family. The multijoints inequality is $|\mathcal J|\leq C^{\mathrm{mult}}_3(|\mathcal L_1||\mathcal L_2||\mathcal L_3|)^{1/2}$. Sharp three-dimensional multijoints constant conjecture. The inequality holds with $C^{\mathrm{mult}}_3=\sqrt{2}$. The source motivates this value using a blow-up of a $K_4$ edge-coloring, but does not establish the bound.