I am asking for references and discussions of statements of the form

> Every bounded hypercover can be refined by an ordinary cover

By "bounded" I mean "finite height". E.g., are there conditions for a site making this statement true? 

My impression is that the statement is true at least for paracompact topological spaces with the Grothendieck topology of open sets, and that, for example, Lemma 7.2.3.5 in 

<cite authors="Lurie, Jacob">_Lurie, Jacob_, [**Higher topos theory**](http://dx.doi.org/10.1515/9781400830558), Annals of Mathematics Studies 170. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14049-0/pbk; 978-0-691-14048-3/hbk). xv, 925&nbsp;p. (2009). [ZBL1175.18001](https://zbmath.org/?q=an:1175.18001).</cite>

can be used to prove this (though some kind of induction would be needed). Has this been proved anywhere explicitly?

My question is related to the question how descent w.r.t. covers differs from descent w.r.t. hypercovers, which has been discussed lately in https://mathoverflow.net/questions/87427/necessity-of-hypercovers-for-sheaf-condition-for-simplicial-sheaves/87446#87446, in https://mathoverflow.net/questions/338618/when-is-the-localization-of-all-hypercovers-equivalent-to-that-of-%c4%8cech-covers, or in https://mathoverflow.net/questions/355762/descent-implies-hyperdescent. In particular, I am aware of the Appendix A in 

<cite authors="Dugger, Daniel; Hollander, Sharon; Isaksen, Daniel C.">_Dugger, Daniel; Hollander, Sharon; Isaksen, Daniel C._, [**Hypercovers and simplicial presheaves**](http://dx.doi.org/10.1017/S0305004103007175), Math. Proc. Camb. Philos. Soc. 136, No. 1, 9-51 (2004). [ZBL1045.55007](https://zbmath.org/?q=an:1045.55007).</cite>

There, Theorem A.10 states that descent w.r.t. to bounded hypercovers is equivalent to descent w.r.t. covers, without any conditions on the site. But I am interested in the more special question about refinement of bounded hypercovers, which, I guess, would then imply the statement about descent.