On the telescope conjecture for module categories

In [H. Krause, O. Solberg, Applications of cotorsion pairs, J. London Math. Soc. 68 (2003) 631–650], the Telescope Conjecture was formulated for the module category image of an artin algebra R as follows: “If image is a complete hereditary cotorsion pair in image with image and image closed under direct limits, then image”. We extend this conjecture to arbitrary rings R, and show that it holds true if and only if the cotorsion pair image is of finite type. Then we prove the conjecture in the case when R is right noetherian and image has bounded injective dimension (thus, in particular, when image is any cotilting cotorsion pair). We also focus on the assumptions that image and image are closed under direct limits and on related closure properties, and detect several asymmetries in the properties of image and image.