Nonexistence of the specified planar semi-cover of $K_{1,2,2,2}$

Let $K_{1,2,2,2}$ be the complete multipartite graph with parts of sizes $1,2,2,2$. A planar semi-cover $G'$ is the planar graph notion defined in the source, and let $H$ be the subgraph required to satisfy the conditions listed in Lemma

. **Nonexistence conjecture.** There is no planar semi-cover $G'$ of $K_{1,2, 2, 2}$ having the properties listed in Lemma

.

The source presents this as a strengthening of Negami's conjecture and aims to prove it as part of the analysis of possible planar covers of $K_{1,2,2,2}$. The supplied text does not establish the claim, so its resolution is left open.