We analyze the parametric space of the constrained minimal supersymmetric standard model with $\mu >0$ supplemented by a generalized asymptotic Yukawa coupling quasiunification condition which yields acceptable masses for the fermions of the third family. We impose constraints from the cold dark matter abundance in the Universe and its direct-detection experiments, the $B$ physics, as well as the masses of the sparticles and the lightest neutral $CP$-even Higgs boson. Fixing the mass of the latter to its central value from the LHC and taking $40\lesssim \tan \beta \lesssim 50$, we find a relatively wide allowed parameter space with $-11\lesssim A_0/M_{1/2}\lesssim 15$ and a mass of the lightest sparticle in the range (0.09–1.1) TeV. This sparticle is possibly detectable by the present cold dark matter direct search experiments. The required fine-tuning for the electroweak symmetry breaking is much milder than the one needed in the neutralino-stau coannihilation region of the same model.