Q1. Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite $T_1$ space is an example, and a more interesting example is the remainder of the Stone-Cech compactification of the integers, see this MSE question and also Parovicenko space.)
There are versions of the above question for LOTS (i.e. linearly ordered topological spaces with the order also called open-interval topology), and for topological groups.
Q2. Is there a compact connected LOTS in which every non-empty $G_\delta$ set has non-empty interior? (I believe the following is a restatement: Is there a compact connected LOTS which is not first-countable at any point?)
Q3. Is there a compact connected Hausdorff topological group in which every non-empty $G_\delta$ set has non-empty interior? (Such a group would necessarily have to be non-metrizable.)
Question 1 was inspired by the following MSE question (in particular by my answer there). Questions 2 and 3 are related and I feel the answers (to all questions) are probably known, in that case would you please provide reference(s).
Edit. Thanks to @ChristianRemling for pointing out that I am interested in examples with at least 2 points. (It follows that such examples, being connected and normal, will have at least continuum many points.)
Edit. The following answer was posted by @Alessandro Vignati (who perhaps could not post it is a comment): "You should have a look at the Stone-Cech reminder of $[0,1)$. I suspect this may be an example of what you're looking for in Q1".
First thank you for the suggested answer. While I agree that the Stone-Cech remainder of $[0,1)$ (or of $[0,\infty)$) is an example to look at, and I had already taken a (brief) look at related papers, I could not deduce the answer, so I posted the above question. Perhaps I am overlooking something obvious, but at any rate unless I see a more specific reference to the suggested result, or a (sketch of) proof, I would not consider the above question answered. Here are a couple of links to papers related to the Stone-Cech remainder $H^*$ of the half-line $H=[0,\infty)$ (indeed $H^*$ is compact and connected, and perhaps satisfies the condition about $G_\delta$ sets I am asking about, but I do not quite see this part, even if I agree it is not something unreasonable to suspect): http://arxiv.org/pdf/math/9805008.pdf and http://arxiv.org/pdf/0708.0838.pdf