Finite-codimension criterion for finite determinacy in positive characteristic

Let $K$ be the ground field, let $R=K[[x_1,\ldots,x_s]]$ with maximal ideal $\mathfrak{m}$, and let $A\in\mathfrak{m}\cdot M_{m,n}$ be a matrix. Let $G$ act on $M_{m,n}$ by the equivalence considered in the paper, and let $\widetilde T_A(GA)$ denote the tangent image at $A$ to the orbit $GA$. Finite $K$-codimension of $\widetilde T_A(GA)$ means that $M_{m,n}/\widetilde T_A(GA)$ is finite-dimensional over $K$. The conjecture. Statement 2 of the cited theorem should hold also when $\operatorname{char}(K)>0$: the condition that there exists an integer $k\geq 0$ such that

$$\mathfrak{m}^{k+2}\cdot M_{m,n}\subset \mathfrak{m}\cdot\widetilde T_A(GA)$$

should be equivalent to $A$ being finitely $G$-determined. In positive characteristic, this asserts in particular that finite codimension of the tangent image is necessary for finite $G$-determinacy. The theorem establishes this equivalence in characteristic zero, while the source explicitly states that its validity in positive characteristic is unknown; the tangent image can be strictly smaller than the tangent space there.