AFAIK the Agda implementations use a closed universe of types and cannot be translated to an implementation of runST in Haskell which doesn't have that restriction. Please correct me if I'm wrong. From a quick reading I can't tell if the new paper makes progress on that front, what do you think?
This information may be out of date. There are now at least two Agda formalisations of runST (by Andrea Vezzosi and András Kovács), and a paper by Timany et al. on the subject of runST has been submitted to ICFP'17.
http://code.haskell.org/~Saizan/ST/ST.agda
http://gist.github.com/AndrasKovacs/07310be00e2a1bb9e94d7c8d...
http://iris-project.org/pdfs/2017-icfp-runST-submission.pdf