Krzyż's conjecture for Taylor coefficients of zero-free disk maps

Let $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ and let $\Omega$ be the set of holomorphic functions $f:\mathbb{D}\to\mathbb{D}\setminus\{0\}$. Write $f(z)=\sum_{k\geq 0}\widehat f(k)z^k$ and let $K_n^\bullet$ denote the corresponding supremum of the modulus of the $n$th Taylor coefficient.

Krzyż's conjecture. For every $f\in\Omega$ and every $n\geq 1$,

$$K_n^\bullet=\sup\{ |\widehat f(n)|:f\in\Omega\}=\frac{2}{e}.$$

Moreover, equality for a given $n$ holds if and only if

$$f(z)=\alpha\exp\!\left(\frac{\zeta z^n-1}{\zeta z^n+1}\right),\qquad |\alpha|=|\zeta|=1.$$

The conjecture is known for $n\leq 5$, while the cases $n\geq 6$ remain open. It is a sharp coefficient problem for zero-free holomorphic self-maps of the disk and is connected with several equivalent Hardy-space extremal conditions.