Weak Zalcman conjecture via the functionals $(C_r)$

Let $S$ be the class of normalized univalent functions in the unit disk, with

$$f(z)=z+\sum_{k=2}^\infty a_k z^k,$$

and, for $r\in[0,1]$, let $(C_r)$ denote the statement

$$|a_n^2-a_{2n-1}|+r|a_{2n-1}|\le (n-1)^2+r(2n-1)$$

for every $f\in S$ and every $n\ge 2$. Second weak version of the Zalcman conjecture. There exists $r\in[0,1]$ such that $(C_r)$ holds.

This is a weaker form of Zalcman's conjecture arising from the corresponding equivalent inequality with a parameter. The source does not establish the existence of such an $r$, so the claim remains open.