The Reduction Conjecture for universal graphs

Let $C$ be a finite or countable graph with no isolated vertices, and decompose it into blocks, its 2-connected components. Let $\tilde C$ be the underlying tree of $C$: its vertices are the blocks of $C$, and two vertices are adjacent when the corresponding blocks share a common vertex. A graph is $C$-free if it contains no copy of $C$, and universality is understood in either the weak or strong sense. Reduction Conjecture. If there is a $C$-free universal graph, in either sense, then there is a $\tilde C$-free universal graph, where $\tilde C$ is the underlying tree of $C$. The conjecture would reduce universality questions for forbidden connected graphs to universality questions for their underlying trees. The supplied text gives no resolution, so the conjecture remains open.