Non-maximal Anosov representations from surface groups to $\mathrm{SO}_0(2,3)$

We prove the representation given by a stable $\alpha_1$-cyclic parabolic $\mathrm{SO}_0(2,3)$-Higgs bundle through the non-Abelian Hodge correspondence is $\{\alpha_2\}$-almost dominated. This is a generalization of Filip's result on weight $3$ variation of Hodge structures and answers a question asked by Collier, Tholozan and Toulisse.