Smallest containing simplex

The problem seems to be still open even for $n=3$:
Weisstein, Eric W. "Tetrahedron Circumscribing."

The paper Parallelotopes of Maximum Volume in a Simplex by Lassak gives the maximum possible volume of a parallelotope in a simplex as $n!/n^n$ times the volume of the simplex. This gives us a bound of $V_n \geq n^n/n!$, which I suspect is tight.

In the paper Minimum area of circumscribed polygons, in Elemente der Mathematik Vol. 28 (1973), Chakerian proved the following:

Any convex body K in Euclidean n-space is contained in a simplex T of volume not more than n^{n-1} times that of K.

Nothing is said there about the extremal cases, so it is possible that the bound is not tight.