Characterization conjecture for arithmetic functions satisfying a quadratic-form equation

Let $k\in\mathbb{N}$ and let $f:\mathbb{N}\to\mathbb{C}$ satisfy

$$f(u^2+kv^2)=f(u)^2+kf(v)^2$$

for all $u,v\in\mathbb{N}$. Characterization conjecture. Exactly one of the following possibilities holds: (1) $f(n)\equiv0$; (2) for every $n\in\mathbb{N}$, $f(n)=n$ if there exist $u,v\in\mathbb{N}$ such that $n=u^2+kv^2$, and $f(n)=\pm n$ otherwise; or (3) for every $n\in\mathbb{N}$, $f(n)=\frac{1}{k+1}$ if there exist $u,v\in\mathbb{N}$ such that $n=u^2+kv^2$, and $f(n)=\pm\frac{1}{k+1}$ otherwise. The conjecture extends the preceding theorem, which establishes the characterization only in the cases treated there; the general validity for all $k\in\mathbb{N}$ remains open.