The spec says:
Once a client receives a PUSH_PROMISE frame and chooses to accept the pushed response, the client SHOULD NOT issue any requests for the promised response until after the promised stream has closed.
Does this explain the quote:
The client should not make any requests to the locally stored promised content until after the promised stream has closed. Presumably this is because the client should wait for the all the promised content to arrive locally; reading before the stream is closed might have incomplete information.