Failure of the Proposed Local Decay Formula for Local Arnold Multiplicities under Twisted Kähler--Ricci Flow

Let $λ(u,x)$ be the local Arnold multiplicity of a quasi-plurisubharmonic function $u$. Di Nezza--Guedj--Lu asked whether every maximal weak solution $φ_t$ of the twisted Kähler--Ricci flow satisfies $λ(φ_t,x)=\max\{λ(φ_0,x)-t,0\}$. We give counterexamples on the Hirzebruch surface $\mathbb F_e=\mat