Must scattered spaces with points $G_\delta$ be locally countable?

Question

By M. E. Gewand, “The Lindelöf degree of scattered spaces and their products,” Journal of the Australian Mathematical Society (Series A) 37 (1984), 98–105, we have:

Theorem. Every Lindelöf scattered $T_3$ space where each point is $G_\delta$ must satisfy $|X|<\omega$.

I think this is a typo and should be $|X|\leq\omega$: a converging sequence is Lindelöf (in fact compact), scattered, with each point $G_\delta$, but is (countably) infinite.

We can see also the following:

Theorem. Every Lindelöf locally countable space must satisfy $|X|\leq\omega$.

Proof. Cover the space with countable open sets, then take a countable subcover. The union is countable and contains the space.

Does the following hold, providing a proof of Gewand's result above?

Conjecture. Every scattered space where each point is $G_\delta$ must be locally countable.

Answer 1

Consider $\mathbb{R}$ with the following topology: a set $U$ is open iff either $0\not\in U$ or $U$ contains an open interval around $0$. This topology is scattered (every point except $0$ is isolated) and every point is $G_\delta$, but it is not locally countable at $0$.