The Milnor-number equality criterion including exceptional vertices

Let $f\in\mathbb{K}[[x,y]]$ be a reduced power series, and let $\mathcal{T}$ be a minimal tree of $f$. Let $\mathcal{V}$ be the set of vertices of $\mathcal{T}$, let $\mathcal{A}_0$ be the distinguished set of additional vertices in the tree construction, and let $N_v$ be the integer attached to each $v\in\mathcal{V}\cup\mathcal{A}_0$. Milnor-number equality criterion. The equality between the Milnor number and the characteristic-independent invariant holds if and only if $\operatorname{Char}\mathbb{K}$ does not divide $N_v$ for all $v\in\mathcal{V}\cup\mathcal{A}_0$. The surrounding discussion identifies this as the equality case of the characteristic-zero formula and the corresponding positive-characteristic inequality, but the supplied text gives no resolution status.