Infinitude conjecture for four simultaneous perfect-square expressions

For $n_1,n_2,n_3\in\mathbb N$ with $n_1,n_2,n_3\geq2$, consider the four integers

$$n_1^2+n_2^2-1,\quad n_1^2+n_3^2-1,\quad n_2^2+n_3^2-1,\quad n_1^2+n_2^2+n_3^2-2.$$

Infinitude conjecture. There exist infinitely many $(n_1,n_2,n_3)\in\mathbb N^3$ with $n_1,n_2,n_3\geq2$ such that all four displayed integers are perfect squares.

The claim asks for infinitely many simultaneous solutions to a system of linear Diophantine conditions on quadratic expressions, related to the construction of analogues of squares and perfect Euler bricks. The supplied context gives computational evidence but no proof or resolution.