Zeng's deflation conjecture for ultrasingular irreducible components

Let $F$ be a polynomial system, let $\textbf{V}(F)$ denote its solution set, and let $V\subseteq\textbf{V}(F)$ be a $k$-dimensional ultrasingular irreducible component. Thus there is a nonempty Zariski open set $U\subseteq V$ consisting of geometric smooth points and a constant $n_0>k$ such that

$$\mathscr{N}\left(J(\textbf{x})\right)=n_0$$

for every $\textbf{x}\in U$. For a solution $\hat{\textbf{x}}$, let $(n_0,n_1,\ldots,n_{i_0})$ be its deflation sequence, and call this sequence regular when its stabilized value equals the local dimension.

Zeng's deflation conjecture. The recursive deflation process terminates after finitely many steps, the deflation sequences of all points $\textbf{x}\in U$ are equal and regular, and for every point $\hat{\textbf{x}}\in V\setminus U$ with deflation sequence $(n_0,n_1,\ldots,n_{i_0})$,

$$\dim_{\hat{\textbf{x}}}\textbf{V}(F)\geq n_{i_0}.$$

The conjecture extends the known regularity of deflation for isolated singular solutions to positive-dimensional ultrasingular components. Its first assertion predicts finite termination and recovery of the component dimension at generic smooth points; its second gives a lower bound for the local dimension at the remaining points. The supplied text gives no resolution, so the status is open.