Monotonicity of mass under the complex-flow construction

Prove or disprove that the mass defined by \(m=\operatorname{tr}|\Omega|\) decreases monotonically under the flow \(h^{-1}dh/dt=-\arg(\Omega)\) for metrized complexes of quiver representations.

Background

The paper extends mass and minimizing flow from quiver representations to complexes of quiver representations using a superconnection-inspired construction. It defines a normal operator Ω\Omega, the mass tr⁡∣Ω∣\operatorname{tr}|\Omega|, and the metric flow h−1dh/dt=−arg⁡(Ω)h^{-1}dh/dt=-\arg(\Omega), but leaves the fundamental monotonicity property unresolved.

References

Does the mass decrease monotonically under the flow~eq:massflowcomplexes?

eq:massflowcomplexes:

m≔tr⁡∣Ω∣,h−1dhdt=−arg⁡(Ω)m\coloneqq \operatorname{tr}|\Omega|, \qquad h^{-1}\frac{dh}{dt}=-\arg(\Omega)

— Towards Categorical Kähler Geometry  (2609.00978 - Haiden et al., 1 Sep 2026) in Section 5.3, subsection “Extension of mass and flow to complexes,” immediately after equation (5.??)