Second-range formula conjecture for the universal bound $g^*(r)$

Let $r\in\mathbb{Z}^{+}$, and suppose there exists $t\in\mathbb{Z}^{+}$ such that

$$t(t+1)\leq r\leq (t+1)^2.$$

Second-range formula conjecture. Then

$$g^*(r)=\frac{2}{t+1}+\frac{1}{t}+\frac{r}{(t+1)^2}-\frac{r}{t(t+1)}.$$

The formula is conjectured as part of the claimed optimality of the constructions in Theorems 7 and 8; no proof or resolution is supplied in the excerpt.