The Hassanzadeh-Nasrollah Nejad-Simis Conjecture on Euler Conductors

We prove the Hassanzadeh-Nasrollah Nejad-Simis conjecture: if $f\in (x_1,\ldots,x_n)^2\subsetneq k[[x_1,\ldots,x_n]]$ has an isolated critical point and $\operatorname{char}k=0$, then its Euler conductor $J_f:f$ is not contained in the Tjurina ideal $(J_f,f)$. More generally, if $A=R/I$ is a nonzero