Optimal strong approximation conjecture for the four-square form

Let $N,m,\lambda_1,\dots,\lambda_4\in\mathbb{Z}$ satisfy

$$N\equiv\sum_{i=1}^4\lambda_i^2\pmod m.$$

Four-square strong approximation conjecture. If $N\gg m^{4+\varepsilon}$, then there exists $(x_1,\dots,x_4)\in\mathbb{Z}^4$ such that

$$\sum_{i=1}^4x_i^2=N,$$

and

$$x_l\equiv\lambda_l\pmod m\qquad(1\leq l\leq4).$$

This is the integral four-variable conjecture whose function-field analogue would yield the sharp upper bound for LPS Ramanujan graph diameters; the source presents it as an optimal strong-approximation conjecture and gives numerical evidence elsewhere.