Near-maximum-degree realizable exponents conjecture for graphs

Let $H$ be a graph, with $v(H)$ vertices, $e(H)$ edges, and maximum degree $\Delta(H)$. A rational number is realizable for $H$ if it occurs as an exponent in the generalized Turán-counting function for $H$. Near-maximum-degree realizable exponents conjecture. Every rational number in the interval

$$\left[v(H)-\frac{e(H)}{\Delta(H)},v(H)\right]$$

is realizable for $H$. This is presented as a slight weakening of the preceding independence-number question. The supplied text gives no resolution of this claim, so its status remains open.