As an amateur I am not quite sure should I post a question on the site for professional mathematicians but if the question is not appropriate for this site you can freely migrate it to MathematicsStackExchange.

I was thinking about (integer) Diophantine $m$-tuples, and (integer) Diophantine $m$-tuple is a set of $m$ positive integers $\{a_1,a_2...,a_m\}$ such that $a_i \cdot a_j +1$ is a perfect square for all $1\leq i <j \leq m$.

It was known for a long time that there exist an infinite number of (integer) Diophantine $4$-tuples and it was conjectured that there does not exist a (integer) Diophantine quintuple but in this paper it is claimed that the problem of quintuples is solved and on this page you can find a lot of information about Diophantine $m$-tuples and various generalizations.

The generalization that I created while I was thinking about (ordinary/classical) Diophantine $m$-tuples is this one:

For every $(k,l) \in \mathbb Z^{2}$ we can look at the (integer) Diophantine problem of $m$-tuples such that $a_i \cdot a_j +ka_i + la_j +1$ is a perfect square for all $1\leq i <j \leq m$ and we see that the classical Diophantine $m$-tuples are just the special case that corresponds to $(k,l)=(0,0)$. Of course that this generalization can be further generalized but at the moment I do not see the need to talk about a generalizations that are not at the heart of the question that I am going to ask and the question is:

Is it true that for every natural number $c \geq2$ there exists at least one pair $(k_c,l_c) \in \mathbb Z^{2}$ and a set of $c$ positive integers $\{a_1,a_2,...,a_c\}$ such that $a_i \cdot a_j +k_ca_i + l_ca_j +1$ is a perfect square for all $1\leq i <j \leq c$?

Is this conjecture true?