The conjectural bounded-distance estimate beyond the square-spacing threshold

Let $a\ge0$, $d\ge1$, and $N\ge1$ be integers, and define

$$\delta=\min_{0\le n\le N,\,m\in\mathbb Z}|a+nd-m^2|.$$

Assume that $a\le\left(\frac{Nd-1}{2}\right)^2$ and $N>d+2\sqrt a$. The conjectural bounded-distance estimate. For every $\epsilon>0$,

$$\delta\ll_\epsilon d^\epsilon.$$

Here the interval contains more squares than residue classes modulo $d$, but additional squares need not produce new residues. This is the complementary range of the preceding distance conjecture and is presented as a consequence of applying it at the threshold.