The Noether inequality for minimal threefolds of general type with small geometric genus

Let $X$ be a minimal projective $3$-fold of general type, and let $p_g(X)$ denote its geometric genus. Assume

$$5\leq p_g(X)\leq 10.$$

Noether inequality. Then

$$K_X^3\geq \frac{4}{3}p_g(X)-\frac{10}{3}.$$

This is the remaining open case of the Noether inequality in dimension $3$: the inequality is known except for finitely many families, while the range $5\leq p_g\leq 10$ is identified in the source as open.