How should one check an instant-equilibration algorithm for conservation?
I would think one of those conditions would always be sufficient. If you have shown that $T'-T=0$ for all $t$ and that $\partial_tT=0$, then  $\partial_tT'=0$ must be true by default. That is $\partial_tT=0$ shows that $T$ is a constant and if $T'-\text{constant}=0$ for all $t$, then $T'$ is also a constant.