Partition regularity of the exceptional linear–quadratic equation

Let $a,b,c$ be non-zero integers, and consider the Diophantine equation

$$a(x_1^2-x_2^2)=by^2+cz.$$

Partition-regularity conjecture. For every choice of non-zero integers $a,b,c$, this equation is partition regular.

This is the sole exceptional case not covered by the paper's linear–quadratic partition-regularity theorem. The authors state that their lack of knowledge of this case is an artefact of their methods and believe that the cited criterion gives the correct characterisation.