Noe–Post conjecture on intersections of generalized Fibonacci sequences

Let $k e \,\ell$ be integers with $\ell>k\geq 2$. The $k$-generalized Fibonacci sequence $F^{(k)}=(F_n^{(k)})_{n\geq -(k-2)}$ has initial values $0,0,\ldots,0,1$ ($k$ terms), with first nonzero term $F_1^{(k)}=1$, and satisfies

$$F_{n+k}^{(k)}=F_{n+k-1}^{(k)}+F_{n+k-2}^{(k)}+\cdots+F_n^{(k)}.$$

Noe–Post conjecture. The Diophantine equation

$$F_m^{(k)}=F_n^{(\ell)},$$

with $n>\ell+1$ and $m>k+1$, has only the solutions

$$(m,n,\ell,k)=(7,6,3,2)\quad\text{and}\quad(12,11,7,3).$$

Equivalently, the intersections of the sequences are expected to consist only of the common initial terms together with $13=F_7^{(2)}=F_6^{(3)}$ and $504=F_{12}^{(3)}=F_{11}^{(7)}$. The paper confirms this conjecture, so the claim is solved.