There is no mni clutter whose core is the projective plane of order 3

In 1998, Lütolf and Margot claimed, in a sidenote, that the projective plane of order $3$, also called the ternary plane, or $PG(2,3$), cannot be the core of any minimally non-ideal clutter. In this paper, we show that, if $\operatorname{\cal C}$ is a minimally non-ideal clutter whose core is the $PG(2,3)$ then $\operatorname{\cal C}$ must be the blocker of the $PG(2,3)$. Since no projective plane of order $n \geq 3$ is minimally non-ideal, the nonexistence of $\operatorname{\cal C}$ is a consequence of the fact that a clutter is minimally non-ideal if and only its blocker is minimally non-ideal.