  The standard AGN-galaxy co-evolutionary scenario predicts a phase of deeply
buried supermassive black hole growth coexisting with a starburst (SB) before
feedback phenomena deplete the cold molecular gas reservoir of the galaxy and
an optically luminous QSO is revealed ('SB-QSO evolutionary sequence'). The aim
of this work is to measure the cold gas reservoir of three highly obscured QSOs
to test if their gas fraction is similar to that of sub-millimeter galaxies
(SMGs), as expected by some models, and place these measurements in the context
of the SB-QSO framework. We target CO(1-0) transition in BzK4892, a Compton
Thick (CT) QSO at z=2.6, CO(1-0) in BzK8608 and CO(2-1) in CDF153, two highly
obscured QSOs at z=2.5 and z=1.5, respectively. For all these targets, we place
3$\sigma$ upper limits on the CO, with $L'_{CO} < (1.5\div 2.8)\times 10^{10}$
K km/s pc$^2$. We also compare the molecular gas conditions of our targets with
those of other systems at z>1, considering normal star forming galaxies and
SMGs, unobscured and obscured AGN from the literature. For the AGN samples, we
provide an updated and (almost) complete collection of targets with CO
follow-up. BzK4892 displays a high star formation efficiency
(SFE$=L_{IR}/L'_{CO}>410$ L$_{\odot}$/(K km s$^{-1}$ pc$^2$)) and a gas
fraction $f_{gas}<0.1$. Less stringent constraints are derived for the other
two targets ($f_{gas}<0.5$ and SFE$>10$). From the comparison with literature
data, we found that a) obscured AGN at z>1 are associated with higher SFE and
lower $f_{gas}$ with respect to star forming galaxies; b) mildly and highly
obscured active galaxies have comparable gas fractions; c) the SFE of CT and
obscured AGN are similar to those of unobscured AGN. Within the SB-QSO
framework, these findings could be consistent with a scenario where feedback
can impact the host galaxy already from the early phases of the SB-QSO
sequence.
