Lin–Zhao stability conjecture for collective strength

Let $\mathbb{K}$ be a field with

$$\operatorname{char}(\mathbb{K})=0\quad\text{or}\quad\operatorname{char}(\mathbb{K})>d,$$

and let $P\in\mathbb{K}[x_1,\dots,x_n]_d$ be a homogeneous polynomial of degree $d$. Here $\operatorname{str}_{\mathbb{K}}(P)$ denotes the strength of $P$, and $\overline{\mathbb{K}}$ is an algebraic closure of $\mathbb{K}$. Lin–Zhao's stability conjecture. One has

$$\operatorname{str}_{\mathbb{K}}(P)\asymp_d\operatorname{str}_{\overline{\mathbb{K}}}(P).$$

The relation $\asymp_d$ denotes linear equivalence with constants depending on $d$. The source states this as a consequence of the partition-rank stability conjecture and notes that the asserted linear bound improves earlier polynomial bounds for infinite perfect fields; the general conjecture remains open there.