Weak Zalcman conjecture via the functionals $(B_t)$

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 $t\in[0,1]$, let $(B_t)$ denote the statement

$$|a_n^2-ta_{2n-1}|\le n^2-t(2n-1)$$

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

This is weaker than Zalcman's conjecture but would still imply the Bieberbach conjecture. The source notes that the case $t=0$ is the Bieberbach conjecture, whereas the existence of a positive admissible $t$ is not known.