Coefficient monotonicity conjecture for the classes $\mathcal{S}_{l,n}^*$

Let $\mathcal{A}$ be the class of normalized analytic functions $f(z)=z+\sum_{m=2}^{\infty}a_mz^m$, and for each integer $n\geq1$ let $\mathcal{S}_{l,n}^*$ consist of the functions $f_n(z)=z+\sum_{m=2}^{\infty}a_{m,n}z^m$ satisfying

$$\frac{zf_n'(z)}{f_n(z)}=1-\log(1+z^n).$$

Coefficient monotonicity conjecture. If $f_n\in\mathcal{S}_{l,n}^*$, then

$$|a_{m,n}|\leq |a_{m,1}|.$$

The conjecture arises from the coefficient pattern obtained by comparing powers in the defining differential equation, and asserts that the absolute value of each coefficient is maximal for the case $n=1$.