Self-intersection conjecture for curves on ruled surfaces

Let $X$ be a ruled surface, let $p_1,\dots,p_r\in X$ be very general points, and let $\mathcal{C}\subset X$ be a reduced irreducible curve. Define $m_i=\operatorname{mult}_{p_i}\mathcal{C}$ for $1\leq i\leq r$, and assume $m_i>0$ for some $i$. Let $X_r$ be the blow-up of $X$ at these points, and write $\widetilde{\mathcal{C}}$ for the strict transform.

Self-intersection conjecture. One has

$$\mathcal{C}^2\geq\sum_{i=1}^r m_i^2-1.$$

Moreover, if equality holds, then if $e\neq0$, $\widetilde{\mathcal{C}}$ is a $(-1)$-curve or $\widetilde{\mathcal{C}}=\widetilde{\Gamma_e}$; while if $e=0$, $\widetilde{\mathcal{C}}$ is a $(-1)$-curve or $\widetilde{\mathcal{C}}=\widetilde{\alpha\Gamma_e}$ for some $\alpha\geq1$. This gives a lower bound for the self-intersection of curves through very general points and, together with the first conjecture, controls negative curves on blow-ups of ruled surfaces; the paper states that the two conjectures are equivalent.