Old question of McLarty on triposes

CAUTION: The Sender of this email is not from within Dalhousie. Dear all, way back in 1998 Colin McLarty raised the following question:

Can the adjoint lifting theorem for triposes (from Andy Pitts's dissertation) be gracefully transported to regular categories (presumably with some supplemental conditions)? If so, the theory of realizability toposes could be radically simplified, among other things (see Japp van Oosten's dissertation).

has there been any progress on or toward this in the last 22 years? Best regards, David David Roberts Webpage: https://ncatlab.org/nlab/show/David+Roberts Blog: https://thehighergeometer.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]