Locally finitely presented Grothendieck categories with a flat generator

A problem raised by Cuadra and Simson in 2007 asks whether a locally finitely presented Grothendieck category with enough flat objects has enough projective objects. We prove that the categories involved in the statement are precisely the locally finitely presented Grothendieck categories with exact product functors. This enables reformulations of the problem that answer the latter in the negative and prove that it is related to a classical problem posed by Miller in 1975 and to the Telescope Conjecture for compactly generated triangulated categories. Our partial affirmative answers imply that any locally finitely presented Grothendieck category whose subcategory of finitely presented objects is Krull--Schmidt has enough flat objects if, and only if, it has a set of finitely generated projective generators.