  For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is
defined by the initial values $0,0,...,0,1$ ($k$ terms) and such that each term
afterwards is the sum of the $k$ preceding terms. In 2005, Noe and Post
conjectured that the only solutions of Diophantine equation
$F_m^{(k)}=F_n^{(\ell)}$, with $\ell>k>1, n>\ell+1, m>k+1$ are
[(m,n,\ell,k)=(7,6,3,2) and (12,11,7,3).] In this paper, we confirm this
conjecture.
