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This recent question received several answers, and GenericHuman's answer and the comments below provide a good synthesis of all the other answers in my opinion.

In this synthesis, only one related question remained unsolved : here it is. Let $\mathbb U$ denote the unit disk in $\mathbb C$, let $E=\lbrace -1,1\rbrace$. For $z=(z_1,z_2, \ldots ,z_n) \in {\mathbb U}^n$ and $\epsilon=(\epsilon_1,\epsilon_2, \ldots ,\epsilon_n) \in E^n$, define

$$ m_{1,n}(z,\epsilon)={\sf max}_{1\leq k \leq n} \Bigg|\sum_{j=1}^k \epsilon_jz_j\Bigg|, $$

$$ m_{2,n}(z)={\sf min}_{\epsilon \in E^n} m_1(z,\epsilon), $$

$$ M_n={\sf sup}_{z\in {\mathbb U}^n} m_{2,n}(z). $$

It has been shown in GenericHuman's answer that $M_n \leq 2$ for all $n$, and $M_2=\sqrt{2}, M_{3} \geq \sqrt{3}$.

Note that $(M_n)$ is an increasing sequence. What is the value of $M_n$ in general ?

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