On the extendability to \(\mathbf {\Pi }_3^0\) ideals and Katětov order

We show that there is a \( \varvec{\Sigma }_4^0\) ${\mathbf{\Sigma }}_4^0$ ideal such that it’s neither extendable to any \( \varvec{\Pi }_3^0\) ${\mathbf{\Pi }}_3^0$ ideal nor above the ideal \( \textrm{Fin}\times \textrm{Fin} \) $\text{Fin}\times \text{Fin}$ in the sense of Katětov order, answering a question from M. Hrušák.