The perfect-power conjecture for reptile simplices

For integers $d\geq 3$ and $k\geq 2$, a $d$-dimensional $k$-reptile simplex is a $d$-dimensional simplex that can be tiled without overlaps by $k$ smaller simplices congruent and similar to it. The perfect-power conjecture. If a $d$-dimensional $k$-reptile simplex exists, then

$$k=m^d$$

for some natural number $m\geq 2$. The paper proves this assertion for $d=3$, while the cases in dimensions $d\geq 4$ remain open.