Proof of the refined Dubrovin conjecture for the Lagrangian Grassmanian $LG(2,4)$

The Dubrovin conjecture predicts a relationship between the monodromy data of the Frobenius manifold associated to the quantum cohomology of a smooth projective variety and the bounded derived category of the same variety. A refinement of this conjecture was given by Cotti, Dubrovin and Guzzetti, which is equivalent to the Gamma conjecture II proposed by Galkin, Golyshev and Iritani. The Gamma conjecture II for quadrics was proved by Hu and Ke. The Lagrangian Grassmanian $LG(2,4)$ is isomorphic to the quadric in ${\mathbb{P}}^4$. In this paper, we give a new proof of the refined Dubrovin conjecture for the Lagrangian Grassmanian $LG(2,4)$ by explicit computation.