Minimality conjecture for reachable and observable linear structured systems

Let $(\cC(s), \cK(s), \cB(s))$ be a linear structured system of order $n$ as given in. A pair $(\cC(s),\cK(s))$ is $\C^n$--observable and a pair $(\cK(s),\cB(s))$ is $\C^n$--reachable when they satisfy the corresponding observability and reachability conditions over $\C^n$.

Minimality conjecture. If $(\cC(s), \cK(s))$ is $\C^n$--observable and $(\cK(s), \cB(s))$ is $\C^n$--reachable, then there is no lower-order realization, constructible via Petrov--Galerkin projection, that realizes the same transfer function.

The preceding result shows that failure of reachability or observability permits a lower-order structured realization of the same transfer function, and that such a realization can be obtained by Petrov--Galerkin projection. The converse for structured systems is left as a conjecture because a concrete proof was not available in the source.