  By the Telescope Conjecture for Module Categories, we mean the following
claim: "Let R be any ring and (A, B) be a hereditary cotorsion pair in Mod-R
with A and B closed under direct limits. Then (A, B) is of finite type."
  We prove a modification of this conjecture with the word 'finite' replaced by
'countable'. We show that a hereditary cotorsion pair (A, B) of modules over an
arbitrary ring R is generated by a set of strongly countably presented modules
provided that B is closed under unions of well-ordered chains. We also
characterize the modules in B and the countably presented modules in A in terms
of morphisms between finitely presented modules, and show that (A, B) is
cogenerated by a single pure-injective module provided that A is closed under
direct limits. Then we move our attention to strong analogies between cotorsion
pairs in module categories and localizing pairs in compactly generated
triangulated categories.
