The simplex–inverted simplex conjecture for volume ratios

Let $n$ be a positive integer, and let a simplex and its inverted simplex be given in $n$-dimensional space. Consider their Minkowski interpolations and define the volume ratio as the ratio of the volume of such an interpolation to the volume of the simplex. Simplex–inverted simplex conjecture. In any dimension $n$, the best volume ratio is generated by the Minkowski interpolation of a simplex and an inverted simplex. This is suggested by the preceding tetrahedral example and by the known two-dimensional case; the assertion for arbitrary dimension remains open.