Conjecture on the curve of maxima of covering constants for Minkowski balls

Let $p$ range over $1<p<\infty$, and consider the curve whose value at $p$ is the maximum covering constant for the Minkowski ball $D_p$. The curve of maxima conjecture. The curve of maxima of covering constants increases from $p=1$ to $p=2$ and decreases from $p=2$ to $p=\infty$. This conjecture concerns the variation of optimal covering constants for Minkowski balls in the plane and identifies $p=2$ as the transition point between the increasing and decreasing regimes; the supplied text gives no resolution.