I have worked on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, to solve completely the equation I have one complicated case and I proved that for this case it is enough to prove the following inequality $$ F_{2m-2}^{(k)}<(F_m^{(k)})^2+1, $$ for all $m\geq 1$ and $k\geq 3$ (or $m>k+1$).

Here, $(F_m^{(k)})_{m\geq -(k-2)}$ is defined by the recurrence $$ F_m^{(k)}=F_{m-1}^{(k)}+F_{m-2}^{(k)}+\cdots + F_{m-k}^{(k)}, $$ with initial values $0,0,\ldots, 0,1=F_1^{(k)}$ ($k-1$ zeroes).

Any suggestion could be very helpful. Thanks in advance!