Farhi's almost-universality conjecture for equal floor-square sums

Let $m$ be an integer with $m\geq 3$, let $n\in\mathbb{N}$, and let $x,y,z\in\mathbb{Z}$. The floor function is defined by

$$\lfloor\theta\rfloor:=\max\{k\leq\theta:k\in\mathbb{Z}\}.$$

Farhi's conjecture. For each integer $m\geq 3$, every natural number $n$ can be represented as

$$n=\left\lfloor\frac{x^2}{m}\right\rfloor+\left\lfloor\frac{y^2}{m}\right\rfloor+\left\lfloor\frac{z^2}{m}\right\rfloor.$$

The restriction $m\geq 3$ is necessary because $\lfloor x^2/2\rfloor$ is even for every $x\in\mathbb{Z}$. The conjecture is known for several small values of $m$, and this paper proves it for every sufficiently large $n$ for each fixed $m\geq 3$; full universality for every $n$ remains open in general.