Infinite five-square progressions over quadratic extensions of quadratic fields

Let $K_1$ be a quadratic extension of $\mathbb{Q}(\sqrt{-2})$ and $K_2$ a quadratic extension of $\mathbb{Q}(\sqrt{2})$. An arithmetic progression of squares is non-constant if its terms are not all equal, and is properly defined over a field if that field is the smallest field over which the progression is defined. Two progressions are equivalent in the sense used by the source. Infinite five-square progression conjecture. There are infinitely many non-constant arithmetic progressions of five squares properly defined over a quadratic extension $K_1$ of $\mathbb{Q}(\sqrt{-2})$, respectively over a quadratic extension $K_2$ of $\mathbb{Q}(\sqrt{2})$. Up to equivalence, in the first case every such progression has the form

$$(a^2,b^2,-2c^2,-m d^2,-2e^2),$$

where $a,b,c,d,e,m\in\mathbb{Z}$ and $m>0$ is square-free, so that $K_1=\mathbb{Q}(\sqrt{-2},\sqrt{-m})$; in the second case every such progression has the form

$$(2a^2,b^2,2c^2,m,e^2),$$

where $a,b,c,e,m\in\mathbb{Z}$ and $m$ is square-free, so that $K_2=\mathbb{Q}(\sqrt{2},\sqrt{m})$.

The claim gives both an infinitude assertion and a classification up to equivalence for five-square arithmetic progressions over the two specified quadratic fields. The supplied parser does not provide evidence that it has been resolved, so its status remains open.