The subdivision conjecture for $K_4$

Let $K_4$ be the complete graph on four vertices, and let a subdivision be a graph obtained by replacing edges by paths. A graph is SPN if it is the sum of a positive semidefinite matrix and a nonnegative matrix.

Subdivision conjecture for $K_4$. Every subdivision of $K_4$ in which all six edges were subdivided, each at least once, is SPN.

The conjecture concerns the remaining subdivisions of $K_4$ in the proposed characterization of SPN graphs. The source states that the case in which each edge is subdivided once is unresolved and does not provide a resolution of the general conjecture.