The nonexistence conjecture for planar covers of the displayed cover of $K_4$

Let $K_4$ be the complete graph on four vertices, and let $\widetilde{K}_4$ be the specific cover of $K_4$ shown in Figure~. A finite planar cover $\widetilde{\widetilde{K}}_4\to\widetilde{K}_4$ is a finite cover whose covering graph embeds in the plane. The composition with $\widetilde{K}_4\to K_4$ is required to be understood as the induced cover of $K_4$.

Nonexistence conjecture for the displayed cover. For every finite planar cover

$$\widetilde{\widetilde{K}}_4\to\widetilde{K}_4,$$

the composition

$$\widetilde{\widetilde{K}}_4\to K_4$$

does not fulfill Properties $\mathcal{V}$ and $\mathcal{E}$.

This conjecture concerns the final example in the paper and is presented as a stronger obstruction for that particular cover. The supplied text gives no resolution status.