Generation conjecture for the centers of compatible Poisson brackets on $\mathfrak{gl}(n)$

The paper considers the compatible Poisson brackets on $\mathfrak{gl}(n)$ discussed in the preceding theorems and remarks, together with the corresponding families of central functions. Let the Poisson tensor in each of these cases be the tensor associated with the relevant bracket.

Generation conjecture. The families of functions listed in the cited theorems and remarks generate the corresponding centers. Equivalently, it suffices to verify that, in each case, the rank of the Poisson tensor is at least $n^2-n$.

This conjecture identifies the displayed central functions as generating families rather than merely elements of the centers. The paper notes that the required rank condition can be checked directly for small $n$, but does not state a general proof.