Conjecture on the tensor-tubal-norm

Let a third-order tensor have a t-SVD with f-diagonal tensor $\boldsymbol{\mathscr{S}}$, and define its tensor-tubal-rank as the number of nonzero diagonal tubes $\boldsymbol{\mathscr{S}}(i,i,:)$. Define the tensor-tubal-norm (TTN) as the sum of the $\ell_2$ norms of these diagonal tubes. Tensor-tubal-norm conjecture. The tensor-tubal-norm is a norm and is the tightest convex relaxation of tensor-tubal-rank. This conjecture concerns a convex surrogate for tensor-tubal-rank, analogous to the nuclear norm as a convex relaxation of matrix rank. The source states that it is based on simulations and tests on a variety of cases, but provides no proof or resolution.