Are Eberlein-Grothendieck scattered spaces $σ$-discrete?

A space $X$ is Eberlein-Grothendieck if $X\subset C_p(K)$ for some compact space $K.$ In this paper we address the problem of whether such a space $X$ is $σ$-discrete whenever it is scattered. We show that if $w(K)\leqω_1$ then $X$ is $σ$-dicrete whenever $X$ has height $ω_1$ and it is locally compact or locally countable. It is also proved that every Lindelöf Čech-complete scattered space is $σ$-compact.

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