Orbital period evolution for all LMXB systems investigated with M₂ = 1.4 M_⊙, M_NS = 1.3 M_⊙ and a magnetic braking index of γ = 2. The first detached system (A) has an initial P_orb = 2.148 days while the widest of the converging systems investigated (B) has an initial P_orb = 2.147 days. Whereas system A leads to an intermediate system (with a final P_orb> 9h), system B does not detach but keeps evolving towards P_min. Therefore, the observed solutions (i.e. RLO detachment and formation of an MSP and a He WD with P_orb = 2−9h) would require a fine-tuning of the initial P_orb to be in a narrow range of less than 2 min. For larger values of γ the situation is less severe (Fig. 2) but still a serious problem, see Sects. 4.2.1, 4.3, and 5.2.