Essential vs Molecular

Given the recent discussion concerning coproducts in toposes (in the thread 'When is Fam(E) a topos?'), I thought it would be a good opportunity to ask if anyone has considered the following problem: To characterize the toposes S such that every essential geometric morphism with codomain S is locally connected. Mat??as Menni. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]