Spherical completeness in the non-Archimedean harmonic norm theorem

Determine whether the spherical completeness assumption on the non-Archimedean field \(K\) can be omitted from the theorem asserting that polystable quiver representations admit harmonic norms and that harmonic norms imply semistability.

Background

The paper proves, under spherical completeness of $K$, that a polystable quiver representation admits a harmonic norm and that the existence of a harmonic norm implies semistability. Spherical completeness is used to ensure that all norms are diagonalizable and that the relevant building is complete. The authors explicitly ask whether this hypothesis is genuinely necessary.