The planar-or-bounded-high-degree structure conjecture for graphs without a topological K_5

Let $K_5$ denote the complete graph on five vertices. A graph is a clique-sum of graphs $G_1,\ldots,G_n$ if it is obtained by repeatedly gluing graphs along cliques. The degree of a vertex is its number of incident edges.

Planar-or-bounded-high-degree conjecture. There exist constants $D$ and $a$ such that every graph $G$ that does not contain $K_5$ as a topological minor can be expressed as a clique-sum of graphs $G_1,\ldots,G_n$ such that, for each $1\leq i\leq n$, either $G_i$ is planar or $G_i$ contains at most $a$ vertices of degree at least $D$.

This conjecture would strengthen the preceding structure theorem by removing the vortices and apex vertices from its planar alternative. Its status is not resolved in the supplied source context.