Curvature-graph conjecture for tight graphs of type $3_n$

Let $G$ be a tight graph of type $3_n$, and let its graph of curvatures be the $4$-vertex graph associated with the four triangular faces of $G$. The three possible graphs of curvatures are the cases displayed in Proposition/Theorem PossibleGraphCurvature.

Curvature-graph conjecture. The graph of curvatures of any tight graph of type $3_n$ is the graph in the first of those three cases.

This conjecture concerns the structure of tight polyhedral graphs with four triangular faces; the source gives the three possible curvature graphs but does not state a resolution of this tighter classification claim.