Conjecture on the only generically identifiable perfect tensor formats

Let a tensor have format $(n_1,\dots,n_d)$, and let

$$R(n_1,\dots,n_d)=\frac{\prod_i n_i}{1+\sum_i(n_i-1)}.$$

Call the format perfect when $R(n_1,\dots,n_d)$ is an integer, and call a format generically identifiable when a general tensor of that format has a unique decomposition as a sum of $R(n_1,\dots,n_d)$ decomposable tensors. The perfect-format identifiability conjecture. The only perfect formats for which a general tensor has a unique decomposition are $(2,k,k)$ for some $k$, $(3,4,5)$, and $(2,2,2,3)$. The formats $(2,k,k)$ are the classically known matrix-pencil cases, identified by Kronecker normal form. The conjecture proposes a complete list of perfect generically identifiable formats, extending the two newly established cases beyond the classical matrix-pencil family; its resolution concerns the uniqueness of tensor decompositions in the balanced perfect-rank setting.