nt.number theory - Smooth proper schemes over Z with points everywhere locally

This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question.

Question. Is there a smooth proper scheme $Xtooperatorname{Spec}(mathbb{Z})$ such that $X(mathbb{Q}_v)neqemptyset$ for every place $v$ of $ mathbb{Q}$ (including $v=infty$), and yet $X(mathbb{Q})=emptyset$ ?

I believe the answer is No. The evidence is flimsy : $X$ cannot be a curve or a torsor under an abelian variety (Abrashkin-Fontaine) or a twisted form of $mathbb{P}_n$.

I haven't gone through a list of smooth projective $mathbb{Q}$-varieties which contradict the Hasse principle to check if any of them has good reduction everywhere, but the chances of such a thing are slim.

Colliot-Thélène and Xu give a systematic treatement of many known examples of quasi-projective $mathbb{Z}$-schemes $Y$ such that $Y(mathbb{Z}_p)neqemptyset$ for every prime $p$ and $Y(mathbb{R})neqemptyset$, but $Y(mathbb{Z})=emptyset$. Some of these schemes might even be smooth over $mathbb{Z}$, but none of them is proper.

Fontaine's letter to Messing (MR1274493) might be of some relevance here.