Odd-primary algebraic transfer injectivity conjecture in ranks at most four

Let $V$ be an elementary abelian group of rank $h$, let $p$ be an odd prime, and let $\mathscr A_p$ be the mod-$p$ Steenrod algebra. The algebraic transfer is the homomorphism

$$Tr_h^{\mathscr A_p}(\mathbb F_p): (\mathbb F_p\otimes_{GL(h,\mathbb F_p)}\operatorname{Ann}_{\overline{\mathscr A_p}}H_*(V;\mathbb F_p))_n\longrightarrow \operatorname{Ext}_{\mathscr A_p}^{h,h+n}(\mathbb F_p,\mathbb F_p).$$

Odd-primary transfer conjecture. The transfer homomorphism $Tr_h^{\mathscr A_p}(\mathbb F_p)$ is one-to-one for any odd prime $p$ and $1\leq h\leq 4$. This extends the known low-rank results at odd primes and is posed as an open problem.