Symmetric energy-momentum tensor

In field theory, there’s no guarantee that the energy-momentum tensor resulting from Noether’s theorem is symmetric. The usual trick to construct a symmetric tensor is to add to the original energy-momentum tensor, the derivative of a function of fields that has three upper indices, and which is anti-symmetric in its first two indices, since the resulting tensor will be conserved.

My question is, although this is a guaranteed way to get a new conserved tensor, but is there a proof in the general case, that there exists at least one function of these functions that we added their derivative, that will make the new energy-momentum tensor symmetric?