The coefficient conjecture for the class of univalent functions with real coefficients

Let $0<\lambda\leq 1$, and let ${\mathcal U}^+(\lambda)$ denote the class of functions with real coefficients considered in the paper. Write $f(z)=z+a_2z^2+a_3z^3+\cdots$. Coefficient conjecture. For every $f\in{\mathcal U}^+(\lambda)$,

$$|a_n|\leq\frac{1-\lambda^n}{1-\lambda}=1+\lambda+\lambda^2+\lambda^3+\cdots+\lambda^n,\qquad n=2,3,4,\ldots.$$

The estimate is known in the broader class ${\mathcal U}(\lambda)$ under the additional subordination condition stated in the source for $n=2,3,4$, while the assertion for all $n$ in ${\mathcal U}^+(\lambda)$ remains open.