Integral ISS conjecture for bilinear systems

Let $\Sigma(A,[0,B_1])$ and $\Sigma(A,[0,B_2])$ denote the two linear systems associated with a bilinear system, and suppose that the bilinear system satisfies the hypothesis of Section 2. Integral ISS conjecture. If both linear systems $\Sigma(A,[0,B_1])$ and $\Sigma(A,[0,B_2])$ are integral input-to-state stable, then the bilinear system is integral input-to-state stable. This conjecture concerns whether integral input-to-state stability of the endpoint linear systems extends to the corresponding bilinear system; in general, the conjecture is open.