Summary: Several recent results have hinted that black hole thermodynamics in loop quantum gravity simplifies if one chooses an imaginary Barbero-Immirzi parameter \(\gamma =i\). This suggests a connection with \(\mathrm{SL}(2,{\mathbb{C}})\) or \(\mathrm{SL}(2,{\mathbb{R}})\) conformal field theories at the ‘boundaries’ formed by spin network edges intersecting the horizon. I present a bit of background regarding the relevant conformal field theories, along with some speculations about how they might be used to count black hole states. I show, in particular, that a set of unproven but plausible assumptions can lead to a boundary conformal field theory whose density of states matches the Bekenstein-Hawking entropy.