  We show that if $k\ge 2$ is an integer and $(F_n^{(k)})_{n\ge 0}$ is the
sequence of $k$-generalized Fibonacci numbers, then there are only finitely
many triples of positive integers $1<a<b<c$ such that $ab+1,~ac+1,~bc+1$ are
all members of $\{F_n^{(k)}: n\ge 1\}$. This generalizes a previous result (cf.
arXiv:1508.07760) where the statement for $k=3$ was proved. The result is
ineffective since it is based on Schmidt's subspace theorem.
