Global stability of a difference equation with maximum

We prove that every positive solution to the difference equationxn=maxA1xn-1α1,A2xn-2α2,…,Akxn-kαk,n∈N0,where k∈N,Ai>0,αi∈(0,1)k∈N,Ai>0,αi∈(0,1), i=1,…,ki=1,…,k, converges to the following quantity maxA11α1+1,…,Ak1αk+1, confirming a quite recent conjecture of interest. We also prove another result on global convergence which concerns some cases when not all αi,i=1,…,kαi,i=1,…,k belong to the interval (0, 1).