The $v_{(2,1)}$-periodicity conjecture for $S/(h,v^4_{(1,0)})$

Let $S/(h,v^4_{(1,0)})$ denote the indicated $C_2$-equivariant complex, let $a$ be the Euler-class element, and let $\widetilde{\bar{\kappa}}$ denote the specified equivariant lift of $\bar{\kappa}$. A $v_{(2,1)}$-self map has the displayed equivariant suspension degree. $v_{(2,1)}$-periodicity conjecture. The complex $S/(h,v^4_{(1,0)})$ has a $v^{32}_{(2,1)}$-self map

$$v^{32}_{(2,1)}: S^{64+128\sigma}/(h,v^4_{(1,0)})\to S/(h,v^4_{(1,0)})$$

such that

$$a^{32}v_{(2,1)}^{32}=\widetilde{\bar{\kappa}}^{8}.$$

This is described as an optimistic conjecture motivated by the classical fact that the minimal $v_2$-self map on $S/(2,v_1^4)$ is a $v_2^{32}$-self map. Its status is not resolved in the given text.