Baily's strength versus Birch-rank conjecture

Let $\mathbb{K}$ be a field, let $d\geq 2$ be an integer, and let $P\in\mathbb{K}[x_1,\dots,x_n]_d$ be a homogeneous polynomial of degree $d$. Suppose that

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

Here $\operatorname{str}_{\mathbb{K}}(P)$ denotes the strength of $P$, and $\operatorname{Brk}(P)$ denotes its Birch rank. Baily's strength versus Birch-rank conjecture. One has

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

The relation $\asymp_d$ denotes linear equivalence with constants depending on $d$. The source says that the partition-rank versus analytic-rank conjecture implies this conjecture over finite fields; its general status is left open.