Large regular simplices contained in a hypercube with a common barycenter

It has been shown that the $n$-dimensional unit hypercube contains an $n$-dimensional regular simplex of edge length $c\sqrt n$ for arbitrary $c<1/2$ if $n$ is sufficiently large (Maehara, Ruzsa and Tokushige, 2009). We prove the same statement holds for some $c>1/2$ even in the special case where a regular simplex has the same barycenter as that of the unit hypercube.