A periodicity-one self-map conjecture for the real-motivic spectrum $\mathcal{Z}_\mathbb{R}$

Let $\mathcal{Z}_\mathbb{R}$ be the real-motivic finite spectrum constructed in the paper, and let $v_{(2,\mathrm{nil})}$ denote a self-map of type $(2,2)$. The suspension $\Sigma^{6,3}\mathcal{Z}_\mathbb{R}$ is related to $\mathcal{Z}_\mathbb{R}$ by a map

$$v:\Sigma^{6,3}\mathcal{Z}_\mathbb{R}\longrightarrow\mathcal{Z}_\mathbb{R}.$$

Periodicity-one self-map conjecture. The spectrum $\mathcal{Z}_\mathbb{R}$ is of type $(2,2)$ and admits a $v_{(2,\mathrm{nil})}$-self-map

$$v:\Sigma^{6,3}\mathcal{Z}_\mathbb{R}\longrightarrow\mathcal{Z}_\mathbb{R}$$

of periodicity $1$. The conjecture is proposed as future work following the construction of $\mathcal{Z}_\mathbb{R}$; the supplied text gives no resolution, so its status remains open.