Bounded complete and dcpo models of T1 spaces

The nice thing about colleagues is that, sometimes, they give me a primer on their latest results.  I would like to talk about a strange result by Dongsheng Zhao and Xiaoyong Xi, which, while accepted for publication, does not seem to be out yet.  (Thanks to D. Zhao for letting me know about this!)  I have already talked about models of topological spaces.  Following earlier results by Zhao, Xi, and Erné, one can show that every T₁ space has a bounded complete, and even algebraic, poset model, and that every T₁ space has a (not bounded complete) dcpo model, but can we have both at the same time?  In other words, does every T₁ space have a bounded complete dcpo model?  Answer (and explanations) in the full post…