The classification of numerically elementary $(-1)$-curves with $b4<2bc_1$

Let $(\delta;\mu_1,\ldots,\mu_r)$ be the numerical data of an elementary $(-1)$-curve in the relevant blow-up of projective space, where $\delta$ is its degree and $\mu_1$ is the largest multiplicity among the assigned points. Such a curve is numerically elementary when its numerical data satisfy the equalities defining an elementary $(-1)$-curve.

Classification conjecture. The only numerically elementary $(-1)$-curves satisfying $\delta<2\mu_1$ are the lines through two points.

The examples preceding the conjecture exhibit numerical sequences satisfying the defining equalities but not corresponding to elementary $(-1)$-curves; together with the stated proposition, they motivate this proposed classification.