Strict monotonicity conjecture for $C(k,p)$

For $0<p<1$, let $C(k,p)$ be the coefficient-estimate constants considered in the paper. Strict monotonicity conjecture. The sequence $C(k,p)$ is strictly increasing. This is presented as a weaker statement that could serve as a starting point for the nonvanishing conjecture. The source does not provide a resolution, so the conjecture remains open.