The non-ellipsoid positive John family conjecture

Let $K$ and $L$ be convex bodies, neither of which is an ellipsoid. Let $(P^*,z^*)$ be the positive John family associated to $K$ and $L$.

Non-ellipsoid positive John family conjecture. The determinant of the matrix component of this family is not identically constant: there is no constant $C$ such that

$$\det P^*=C$$

identically.

The conjecture asserts that the constant-determinant behavior exhibited by ellipsoids in positive John position is unique to that class. Its resolution is not indicated in the supplied text.