Before revision:  We make a conjecture that the sum of squares of the largest s singular valuesof a third order tensor is greater than or equal to the sum of squares of any s entries of that third order tensor. Kilmer and Martin showed that an Eckart-Young theorem holds for the tensor tubal rank of third order tensors. We show that our conjecture is true if and only if another Eckart-Young theorem holds for the T-rank of third order tensors .
After revision:  Some properties of singular values, T-rank and best T-rank of third order tensors .