Geometric local systems on the projective line minus four points

Let $J(m)$ be an $m\times m$ Jordan block with eigenvalue $1$. For $\lambda \in \mathbb{C}\setminus \{0,1\}$, we explicitly construct all rank $2$ local systems of geometric origin on ${\mathbb{P}}^1\setminus \{0,1,\lambda ,\mathrm{\infty }\}$, with local monodromy conjugate to $J(2)$ at $0,1,\lambda$ and conjugate to $-J(2)$ at $\mathrm{\infty }$. The construction relies on Katz’s middle convolution operation. We use our construction to prove two conjectures of Sun-Yang-Zuo (one of which was proven earlier by Lin-Sheng-Wang; the other was proven independently from us by Yang-Zuo).