The bounded-error conjecture for the minimum degree of linearly dependent powers

Let $\Phi_k(d)$ denote the minimum number of linearly dependent $d$th powers of binary quadratic forms in a set of $k$ such powers, as defined in the paper. Here $k$ and $d$ are positive integers.

Bounded-error conjecture. There is a small constant $M$ such that, for all $k$ and $d$,

$$\left\vert \Phi_k(d) - \min_{1 \le i \le k}\left(\frac di + i \right) \right\vert < M.$$

The conjecture proposes that the bounds suggested by the paper's main theorem determine $\Phi_k(d)$ up to a universal additive constant. The preceding discussion indicates that the behavior of these quantities is understood in several cases, but that broader results for larger parameters remain open.