  Motivated by very recent experiments, we consider a scenario "\`a la Bell" in
which two protagonists test the Clauser-Horne-Shimony-Holt (CHSH) inequality
using a photon-pair source based on spontaneous parametric down conversion and
imperfect photon detectors. The conventional wisdom says that (i) if the
detectors have unit efficiency, the CHSH violation can reach its maximum
quantum value $2\sqrt{2}$. To obtain the maximal possible violation, it
suffices that the source emits (ii) maximally entangled photon pairs (iii) in
two well defined single modes. Through a non-perturabive calculation of
non-local correlations, we show that none of these statements are true. By
providing the optimal pump parameters, measurement settings and state structure
for any detection efficiency and dark count probability, our results give the
recipe to close all the loopholes in a Bell test using photon pairs.
