Bruhn–Rautenbach's Fano-plane determinant conjecture

Let $A\in\{0,1\}^{n\times n}$ have at most $3n$ non-zero entries. Bruhn–Rautenbach's conjecture. The determinant of $A$ is at most

$$24^{n/7}.$$

The bound is motivated by the incidence matrix of the Fano plane, whose determinant is $24$, and by disjoint unions of such matrices. The source presents this as an open problem.