Kühn­el–Lassmann minimality conjecture for combinatorial tori

Let $T^d$ be the $d$-dimensional torus, and consider the Kühn­el–Lassmann combinatorial triangulation with $2^{d+1}-1$ vertices.

Kühn­el–Lassmann torus conjecture. The Kühn­el–Lassmann series of combinatorial $d$-tori $T^d$ with $2^{d+1}-1$ vertices is vertex-minimal.

The series gives explicit triangulations for all $d\geq2$. The paper reports alternative $31$-vertex triangulations of $T^4$ and further triangulations in dimension $8$, but no additional $15$-vertex triangulation of $T^3$; the asserted minimality was open in the source.