There is an open problem in this paper: Classes defined by stars and neighbourhood assignments by van Mill and others.

**Problem 4.8.** Is a regular (Tychonoff) star compact space metrizable if it has a $G_\delta$-diagonal?

A topological space $X$ is said to be star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$. More on star compactness see here.

My question is this: Is always the cardinality of such regular (Tychonoff) star compact space less than $2^{\omega_0}$? See the related link here.