Levelness conjecture for four general uniform powers in three variables

Let $R=k[x,y,z]$ and let

$$I=(L_1^k,\dots,L_4^k),$$

where $L_1,\dots,L_4$ are general linear forms. Levelness conjecture. The quotient $R/I$ is level with Cohen–Macaulay type $k$. This predicts a precise socle structure for the Artinian quotient arising from four general uniform powers; the supplied text does not indicate whether the conjecture has been resolved.