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The logarithmic coefficients $\\gamma_n$ of $f$ are defined by the formula $\\log(f(z)/z)=2\\sum_{n=1}^\\infty \\gamma_nz^n$. In a recent paper, the present authors proposed a conjecture that if $f\\in {\\mathcal U}(\\lambda)$ for some $0<\\lambda \\leq 1$, then $|a_n|\\leq \\sum_{k=0}^{n-1}\\lambda ^k$ for $n\\geq 2$ and provided a new proof for the case $n=2$. 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