General higher-even-order hyper-Airy propagation conjecture

Establish for every even-order dispersive Schrödinger equation $i\partial_tu=(-1)^n\partial_x^{2n}u$ with initial data given by a higher-order Airy function satisfying $\mathrm{Ai}_{2n}^{(2n)}(x)-x\mathrm{Ai}_{2n}(x)=0$ that the wave packet accelerates along a caustic with $x\sim t^{2n}$, disperses along the caustic at rate $O(t^{-(2n-2)/3})$, and has a principal lobe near $x\approx0$ that decays at rate $O(t^{-1/(2n)})$, yielding uniform amplitude decay of order $O(t^{-1/(2n)})$.

Background

The paper numerically examines the sixth-order case and observes an accelerating caustic, dispersive decay along that caustic, and a slowly dispersing principal lobe near the origin. It then proposes an extension to arbitrary even-order dispersion using a formal asymptotic analysis modeled on the quartic case.

The claimed scaling laws concern the caustic trajectory, the decay along the accelerating structure, and the dominant decay rate near the origin. Because the authors explicitly describe the general statement as a conjecture rather than establishing it rigorously for all even orders, it qualifies as an unresolved conjectural problem.