Following up on the ideas in @Nick's answer & comments, you can see how wide the bandwidth needs to be to just flatten out the secondary mode:
Take this kernel density estimate as the proximal null—the distribution closest to the data yet still consistent with the null hypothesis that it's a sample from a unimodal population— & simulate from it. In the simulated samples the secondary mode doesn't often look so distinct, & you needn't widen the bandwidth as much to flatten it out.
Formalizing this approach leads to the test given in Silverman (1981), "Using kernel density estimates to investigate modality", JRSS B, 43, 1.