An extremal property of convex hexagons

The following conjecture is discussed: if $K$ is a plane convex figure and $T$ is a triangle of maximal area contained in $K$, then $K$ is contained in $\sqrt5T$. It is shown that it suffices to check the conjecture in the case where $K$ is a convex hexagon, but the conjecture is proved only in the case where $K$ is a pentagon. Bibl. – 2 titles.