Conway's $99$-graph is a strongly regular graph with parameters $(v,k,λ,μ) = (99,14,1,2)$ and whose existence has long been an open problem. We use both a sum-of-squares program and the fact that a strongly regular graph with $λ= 1, μ= 2$ has no diamond or $K_{2,3}$ subgraph to derive a loose upper bound for the number of triangles that such a graph could contain. This bound is less than the number of triangles that the graph's spectrum asserts, implying that it does not exist.

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