Conjecture on polynomial asymptotic structures of PT-symmetric vector rational rogue waves

Let $\mathcal{PT}$-symmetric vector rational rogue waves be solutions of the multi-component nonlinear Schrödinger equations, and let their asymptotic structures refer to the patterns determined by the multiple-root configurations of associated governing polynomials. Polynomial asymptotic-structure conjecture. The asymptotic structures of $\mathcal{PT}$-symmetric vector rational rogue waves are related to some real-coefficient polynomials. The conjecture is motivated by the asymptotic analysis of representative vector rational rogue waves, where the governing polynomial has real coefficients and its multiple-root types determine the observed structures; the precise general relationship remains open.