Erdős Problem #35 — Sharper Schnirelmann density gain from adding a basis of order $k$

The sequence B is called a basis of order $k$, if every integer is the sum of $k$ or fewer $b$-s. Let $d_s(\mathrm{A}) = \alpha$ then I proved that

$$d_s(\mathrm{A} + \mathrm{B}) \geqslant \alpha + \frac{\alpha(1-\alpha)}{2k} . \tag{12}$$

Thus every base is an essential component. Several authors improved (12) in various ways. I conjectured that

$$d_s(\mathrm{A} + \mathrm{B}) \geqslant \alpha + \frac{\alpha(1-\alpha)}{k} . \tag{13}$$