This is a continuation of the question about Minimal $T_0$-spaces .

Let $X\neq \emptyset$ be a set and let $\text{Top}(X)$ denote the lattice of all topologies on $X$ and let $\tau\in\text{Top}(X)$.

Do we have $$\tau = \bigcap\{\sigma \in \text{Top}(X): \sigma \text{ is } T_0 \text{ and } \sigma \supseteq \tau\}?$$