Must a Suslin line be star countable?

1. A Suslin line is defined to be a nonseparable ccc linearly ordered space.

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

My question is this:

Must a Suslin line be star countable?