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

Extensions of continuous functions in $\beta\mathbb{N}$, Bull. Amer. Math. Soc. 66 (1960) 376–381. $\endgroup$ – Ramiro de la Vega Apr 6 '15 at 15:22