The licci classification conjecture for $(c,d,t)$ triplets

Fix integers $c\geq 3$, $d\geq 1$, and $t\geq 1$. For a perfect ideal, the codimension, deviation, and type are respectively $c$, $d$, and $t$; associate to the triplet the diagram $T_{c-1,d+1,t+1}$. The licci classification conjecture. If $T_{c-1,d+1,t+1}$ is Dynkin of type ADE, then every perfect ideal of codimension $c$, deviation $d$, and type $t$ is licci. If the diagram is not Dynkin, then there exist perfect ideals with codimension $c$, deviation $d$, and type $t$ that are not licci. The conjecture summarizes the expected classification of licci perfect ideals by the associated diagram. The surrounding text describes the result as conjectural and records that the general question remains open.