A counterexample to the Bollobas-Riordan conjectures on sparse graph limits

Bollobas and Riordan, in their paper 'Metrics for sparse graphs', proposed a number of provocative conjectures extending central results of quasirandom graphs and graph limits to sparse graphs. We refute these conjectures by exhibiting a sequence of graphs with convergent normalized subgraph densities (and pseudorandom C-4-counts), but with no limit expressible as a kernel.