Existence of an even dimension with minimal absorption index equal to n for the cube

Determine whether there exists an even positive integer n such that the minimal absorption index of the unit cube Q_n by an inner nondegenerate simplex equals n; that is, ascertain whether there exists even n with \xi_n(Q_n)=n.

Background

For the unit cube Q_n, the minimal absorption index \xi_n(Q_n) lies in [n, n+1), and exact values are known for n=2,5,9 and many n with Hadamard matrices, where typically \xi_n(Q_n)=n.

The paper emphasizes that among even n, only n=2 currently has a fully determined value, and it is unknown if any even n satisfy \xi_n(Q_n)=n.