Segre's version of the SHGH conjecture for general fat points

Let $V=[R]_j$ be the vector space of degree-$j$ forms in three variables, let $\mathcal{L}_j$ be its projectivization, and let $X=m_1P_1+\cdots+m_rP_r$ be the fat point scheme supported at general points $P_1,\ldots,P_r$, with ideal

$$I_X=I_{P_1}^{m_1}\cap\cdots\cap I_{P_r}^{m_r}.$$

The scheme $X$ fails to impose the expected number of conditions on $V$ when

$$\dim_K[I_X]_j>\max\left\{0,\binom{j+2}{2}-\sum_i\binom{m_i+1}{2}\right\}.$$

Segre's version of the SHGH conjecture. For $X=m_1P_1+\cdots+m_rP_r$ with general points $P_i$, $X$ fails to impose the expected number of conditions on $V$ only if $[I_X]_j\ne0$ but the base locus of $[I_X]_j$ contains a non-reduced curve. This is presented as a conjectural necessary criterion within the broader open problem of classifying the failures of the expected-condition count for general fat points.