The Dimension Conjecture for fewer differential equations than variables

Let $(K,\partial)$ be a differential field of characteristic zero, let $u_1,\ldots,u_m\in K\lbrace x_1,\ldots,x_n\rbrace$, and let

$$\Sigma=\operatorname{Spec}K\lbrace x_1,\ldots,x_n\rbrace/[u_1,\ldots,u_m].$$

Dimension Conjecture. If $m<n$, then every irreducible component $\Sigma_1$ of $\Sigma$ satisfies

$$\dim_K^{\partial}(\Sigma_1)\geq n-m.$$

This is presented as a stronger formulation of the assertion that fewer equations than variables force positive differential dimension. The source states that the Dimension Conjecture, including this formulation and the other formulations listed in the paper, is open.