Singer-type upper-bound conjecture for optimal Golomb rulers

Let $G(k)$ be the length of an optimal Golomb ruler with $k$ marks. Singer-type upper-bound conjecture. For every integer $k\geq6$,

$$G(k+2)<k^2+k.$$

The source derives this as a consequence of the conjecture $H(I,I+2)=I(I+2)$ and notes that it would imply the stronger range-shifted form of the Erdős conjecture $G(k)<k^2$; the statement itself remains conditional and unproved.