In  Jim Cushing and Shandelle Henson published two conjectures (see Section 3) related to the Beverton-Holt difference equation (with growth parameter exceeding one), which said that the B-H equation with periodically varying coefficients (a) will have a globally asymptotically stable periodic solution and (b) the average of the state variable along the periodic orbit will be strictly less than the average of the carrying capacities of the individual maps. They had previously  proved both statements for period 2.
Taylor & Francis
Elaydi, S., & Sacker, R.J. (2005). Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures. Journal of Difference Equations and Applications, 11(4-5), 337-346. doi:10.1080/10236190412331335418
Journal of Difference Equations and Applications