  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,
$~i=\overline{1,k}$.
