Sylvester-type quaternion matrix equations with arbitrary equations and arbitrary unknowns

In this paper, we prove a conjecture which was presented in a recent paper [Linear Algebra Appl. 2016; 496: 549--593]. We derive some practical necessary and sufficient conditions for the existence of a solution to a system of coupled two-sided Sylvester-type quaternion matrix equations with arbitrary equations and arbitrary unknowns $A_{i}X_{i}B_{i}+C_{i}X_{i+1}D_{i}=E_{i},~i=\overline{1,k}$. As an application, we give some practical necessary and sufficient conditions for the existence of an $\eta$-Hermitian solution to the system of quaternion matrix equations $A_{i}X_{i}A^{\eta*}_{i}+C_{i}X_{i+1}C^{\eta*}_{i}=E_{i}$ in terms of ranks, $~i=\overline{1,k}$.