1. A space $X$ is said to be star-Menger if for every sequence $(\mathcal{U}_n)$ of open covers of $X$ there exists a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a finite subset of $\mathcal{U}_n$ and $\cup_{n\in\mathbb N}\{St(V,\mathcal{U}_n) : V\in\mathcal{V}_n\}$ is an open cover of $X$.
2. A space $X$ is said to be star-$K$-Menger if for every sequence $(\mathcal{U}_n)$ of open covers of $X$ there exists a sequence $(K_n)$ of compact subsets of $X$ such that $\{St(K_n,\mathcal{U}_n) : n\in\mathbb N\}$ is an open cover of $X$.

It is well known that every star-$K$-Menger space is star-Menger. Give an example of a star-Menger space which is not star-$K$-Menger.