What is the physical significance of the vector field term $X_{\nu}$ in the improved Noether current $T^{\mu\nu}X_{\nu}$?

In Pedro Lauridsen Ribeiro's answer to deriving the improved stress-energy tensor using the improved Noether current, the variational equation for the improved stress-energy tensor is given by: $$$$2\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}\nabla_\mu X_\nu+\nabla_\mu(T^{\mu\nu}X_\nu)=\left(2\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}+T^{\mu\nu}\right)\nabla_\mu X_\nu+X_\nu\nabla_\mu T^{\mu\nu}=0.\$$$$ Here $$X$$ is an arbitrary vector field $$X$$ on $$M$$ (i.e. an infinitesimal diffeomorphism).

In the above example, where the covariant derivative is defined with the levi-civita connection and the stress tensor is conserved, $$X_{\nu}$$ and its derivative act trivially and do not have physical significance. We see that $$X_{\nu}$$ is contracted with the conserved divergence of the stress tensor in the last term. In a hypothetical world, if the stress tensor was not conserved what would the physical meaning of $$X_{\nu}$$ be?

Whatever I understand most of the cases, we do not really know whether the $$T^{\mu\nu}$$ is actually symmetric or not, but, it should be conserved covariantly because improvement terms are in general, covariantly conserved. But they can alter the symmetry property of the stress energy tensor.
• I am not asking about $T^{\mu\nu}$, I am asking about $X_{\nu}$. To get intuition of $X_{\nu}$, I am asking what its physical significance would be if it were not arbitrary and not independently defined from its derivative. This would happen in a made up world where $T^{\mu\nu}$ is not conserved and/or using a different connection.