Relative K-stability of the limiting Wang–Zhou pentagon

Determine whether the pentagon with vertices \((-1,-\alpha-2k), (-1,\alpha+4), (0,\alpha+4), (1,\alpha), (1,-\alpha)\) is relatively K-stable for the relevant parameters, beyond the result that it is not destabilized by the specific test functions \(u_t=\max(y-\alpha+kt,0)\).

Background

The pentagon is the limiting polygon obtained from Wang and Zhou’s octagon family as the parameters $\epsilon_i$ tend to zero. Wang and Zhou claimed that it is destabilized by the simple piecewise-linear functions $u_t=\max(y-\alpha+kt,0)$.

The paper disproves that specific claim by showing that the Donaldson–Futaki invariant of $u_t$ is asymptotically positive. However, ruling out this particular class of test functions does not establish relative K-stability, and the authors explicitly leave the full stability question unresolved.