  The recent experimental hints for a relatively heavy Higgs with a mass in the
range 119-125 GeV favour supersymmetric scenarios with a large mixing in the
stop mass matrix. It has been shown that this is possible in the constrained
Minimal Supersymmetric Standard Model (CMSSM), but only for a very specific
relation between the trilinear parameter and the soft scalar mass, favouring A
~ -2m for a relatively light spectrum, and sizable values of tan(beta). We
describe here a string-derived scheme in which the first condition is automatic
and the second arises as a consequence of imposing radiative EW symmetry
breaking and viable neutralino dark matter in agreement with WMAP constraints.
More specifically, we consider modulus dominated SUSY-breaking in Type II
string compactifications and show that it leads to a very predictive CMSSM-like
scheme, with small departures due to background fluxes. Imposing the above
constraints leaves only one free parameter, which corresponds to an overall
scale. We show that in this construction $A=-3\sqrt{2}m\approx -2m$ and in the
allowed parameter space tan(beta)=38-41, leading to 119 GeV < m_h < 125 GeV. We
determine the detectability of this model and show that it could start being
probed by the LHC at 7(8) TeV with a luminosity of 5(2) fb^-1, and the whole
parameter space would be accessible for 14 TeV and 25 fb^-1. Furthermore, this
scenario can host a long-lived stau with the right properties to lead to
catalyzed BBN. We finally argue that anthropic arguments could favour the
highest value for the Higgs mass that is compatible with neutralino dark
matter, i.e., m_h~125 GeV.
