  We provide several general upper bounds on the rate of a key secure against a
quantum adversary in the device-independent conference key agreement (DI-CKA)
scenario. They include bounds by reduced entanglement measures and those based
on multipartite secrecy monotones such as a multipartite squashed
entanglement-based measure, which we refer to as reduced c-squashed
entanglement. We compare the latter bound with the known lower bound for the
protocol of conference key distillation based on the parity
Clauser-Horne-Shimony-Holt game. We also show that the gap between the DI-CKA
rate and the device-dependent rate is inherited from the bipartite gap between
device-independent and device-dependent key rates, giving examples that exhibit
the strict gap.
