The classification conjecture for extensions of the pair {k,k+1}

Let $k$ be a positive integer, and let $\{k,k+1,c,d\}$ be a $D(-k)$-quadruple with $c<d$, where a $D(-k)$-quadruple is a set of four positive integers whose pairwise products minus $k$ are perfect squares.

Classification conjecture. One has

$$c=1,\qquad d=4k+1,$$

and, in this case, $3k+1$ must be a square.

The conjecture is presented as a problem for future research and is stated to be more difficult for general $k$ than related cases already considered. It would classify all such extensions of the pair $\{k,k+1\}$ subject to $c<d$.