Average density evolution of the post-AGB stars for the initial mass of 3.1 M_⊙ (shown in panel a) and 10 M_⊙ (shown in panel b). The shading regions represent the conditions to produce QPEs given in Table 2, which can last for several times 10³ years and 10⁴ years for M₂ = 0.6 and M₂ = 1.1, respectively. The age and average density corresponding to Fig. 5 are represented by red stars, and the stellar properties are listed in Table 3.