1. Let $d$ be the common difference of the arithmetic sequence of the interior angles of the convex hexagon. Assume without loss of generality (WLOG) that $d \ge 0$ (if $d < 0$, we can consider the absolute value of $d$).

2. The sum of the interior angles of a convex hexagon is given by:
$$180^\circ \times (6 - 2) = 720^\circ$$
Since there are six angles, the average measure of each angle is:
$$\frac{720^\circ}{6} = 120^\circ$$

3. The angles form an arithmetic sequence, so we can denote the angles as:
$$120^\circ - \frac{5d}{2}, 120^\circ - \frac{3d}{2}, 120^\circ - \frac{d}{2}, 120^\circ + \frac{d}{2}, 120^\circ + \frac{3d}{2}, 120^\circ + \frac{5d}{2}$$

4. For the hexagon to be convex, all interior angles must be between $0^\circ$ and $180^\circ$. Therefore, the smallest angle must be greater than $0^\circ$ and the largest angle must be less than $180^\circ$.

5. The smallest angle is:
$$120^\circ - \frac{5d}{2} > 0^\circ$$
Solving for $d$:
$$120^\circ > \frac{5d}{2}$$
$$240^\circ > 5d$$
$$48^\circ > d$$
$$d < 48^\circ$$

6. The largest angle is:
$$120^\circ + \frac{5d}{2} < 180^\circ$$
Solving for $d$:
$$120^\circ + \frac{5d}{2} < 180^\circ$$
$$\frac{5d}{2} < 60^\circ$$
$$5d < 120^\circ$$
$$d < 24^\circ$$

7. Combining both conditions, we get:
$$d < 24^\circ$$

8. Therefore, the greatest possible value of $D$ is:
$$\boxed{24}$$