Non-compact and maximal non-$T_2$ Is there a space $(X,\tau)$ that is not compact, not $T_2$, but for every topology $\tau'\supseteq \tau$ with $\tau'\neq\tau$ the space $(X,\tau')$ is $T

Non-compact and maximal non-$T_2$ Is there a space $(X,\tau)$ that is not compact, not $T_2$, but for every topology $\tau'\supseteq \tau$ with $\tau'\neq\tau$ the space $(X,\tau')$ is $T_2$?

Let $X$ be an infinite set and fix $a\
eq b$ in $X$. Let $\tau$ be the topology generated by the singletons $\\{x\\}$ for $x\
eq a$ and the set $\\{a,b\\}$. So every set $U\subset X$ is open, provided that $a\in U\to b\in U$. This is not compact and not $T_2$, since we cannot separate $a$ from $b$. But the only larger topology is the discrete topology, which is $T_2$.