The Solidity Conjecture for universal graphs with a forbidden graph

Let $C$ be a finite or countable graph with no isolated vertices. Decompose $C$ into its blocks, where each block is a 2-connected component, and let $\tilde C$ be the underlying forest whose vertices are the blocks of $C$, with adjacency determined by sharing a vertex. A graph is $C$-free if it contains no copy of $C$, and a universal graph may be universal either in the weak or strong sense. Solidity Conjecture. If there is a $C$-free universal graph, in either the weak or strong sense, then every block of $C$ is complete. This conjecture asserts that the existence of a universal graph avoiding one finite connected constraint forces the constraint's block structure to be solid. The supplied text gives no resolution, so the conjecture remains open.