Nonexistence of planar semi-covers of $K_{2,2,2,1}$ under specified conditions

A planar semi-cover of $K_{2,2,2,1}$ is a planar graph $G'$ equipped with the projection structure described in the paper. Suppose $G'$ satisfies the following conditions: if $H\subset G'$ is the full preimage of the subgraph $K_4\subset K_{2,2,2,1}$ on vertices $0,-1,-2,-3$, then $H$ is connected, the restriction of the projection map to $H$ is a genuine cover, and the outer boundary $C_e$ of $G'$ is a cycle of $H$; all $3$-cycles of $G'$ are facial; all cycles of the relevant graph $G$ covering the cycles $(1,2,3)$ and $(-1,-2,-3)$ of $K_{2,2,2,1}$ are $3$-cycles; and paths covering other octahedral $3$-cycles may start and end on the boundary of $G'$. Nonexistence conjecture. There is no planar semi-cover $G'$ of $K_{2,2,2,1}$ satisfying these conditions. This is presented as a strengthening of the paper's earlier conjecture concerning planar covers, and is intended to eliminate an infinite class of possible components in the proof of the main theorem.