The path-linked triangles 3-colourability conjecture

Let $H$ be a graph which is a disjoint union of triangles and paths of length at most $12$, and let $G$ be obtained from $H$ by gluing on vertex-disjoint triangles. For two triangles $T,T'$ in $H$, say they are path-linked if there exists a glued-in triangle $Y$ with $T\cap Y,T'\cap Y\neq\emptyset$ and the third vertex of $Y$ lies on a path in $G$ of length at least two. Path-linked triangles conjecture. If every $H$-triangle in $G$ is path-linked to at most one other $H$-triangle, then $G$ is 3-colourable.

The restrictions are intended to avoid the known obstructions arising from more general unions of paths and triangles; the case where all $H$-paths have length at most $2$ follows from an auxiliary bipartite-graph argument. The conjecture is presented as approachable, but remains open in the source.