Complex discrepancy conjecture for trees

Let $T$ be a tree with $\ell$ leaves, let $\mathcal{T}$ denote the family of trees used in the discrepancy definition, and let $\mathcal{D}^1(T,\mathcal{T})$ denote the one-dimensional, or complex, discrepancy. Write $\Delta(T)$ for the maximum degree of $T$. Complex discrepancy conjecture. For every tree $T$ with $\ell$ leaves,

$$\mathcal{D}^1(T,\mathcal{T})\ge \frac{1}{2\sin\left(\frac{\pi}{2\ell}\right)}=\left(1+o(1)\right)\frac{\ell}{\pi}.$$

If, in addition, $\Delta(T)=\omega(1)$, then

$$\mathcal{D}^1(T,\mathcal{T})=\left(1+o(1)\right)\frac{\ell}{\pi}.$$

The preceding theorem proves the weaker universal lower bound $\mathcal{D}^1(T,\mathcal{T})\ge \ell/\pi$; the conjecture proposes the sharp trigonometric bound and asymptotic equality for trees of unbounded maximum degree.