Let $X\neq \emptyset$ be a set and let ${\cal J} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ be a collection of non-empty subsets of $X$. We say that a topology $\tau$ on $X$ is ${\cal J}$-*compatible* if for every $J\in {\cal J}$ there is a continuous surjective map $f:X\to J$, where $J$ carries the subset topology inherited from $(X,\tau)$.

Let $\text{Cptb}(X, {\cal J})$ denote the collection of ${\cal J}$-compatible topologies, ordered by $\subseteq$.

Note that ${\cal P}(X)$ is always the greatest element of $\text{Cptb}(X,{\cal J})$, and the indiscrete topology $\{\emptyset, X\}$ is always the smallest element.

If $\tau\in \text{Cptb}(X, {\cal J})$ and $\tau\neq{\cal P}(X)$, is there a minimal element in $\text{Cptb}(X, {\cal J})$ properly containing $\tau$ (= "above" $\tau$, in the poset we are considering)?