The proof of a conjecture concerning the intersection of k-generalized Fibonacci sequences

For $k\ge 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)}{)}_n$ is defined by the initial values $0,0,\mathrm{.}\mathrm{.}\mathrm{.},0,1$ ($k$ 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 $F_m^{(k)}=F_n^{(\mathrm{\ell })}$, with $\mathrm{\ell }>k>1,n>\mathrm{\ell }+1,m>k+1$ are [(m,n,\ell,k)=(7,6,3,2) and (12,11,7,3).] In this paper, we confirm this conjecture.