Will proof of work used for anything after Coordicide goes live?
Yes, PoW will still be used for spam protection and rate control. The adaptive difficulty will depend on the current throughput of the network and the reputation of the node (ie. mana).
It seems like it wont anymore. In the abstract of https://arxiv.org/abs/2005.07778 it says,
In this work, we consider a novel IoT-friendly DLT architecture based on directed acyclic graphs, rather than a blockchain, and with a reputation system in the place of Proof of Work (PoW).
Similarly in their specification, they mention for the congestion control a,
A TCP-inspired algorithm for decentralized rate setting to efficiently utilize the available bandwidth while preventing large delays.
With no mention of POW. As you probably remember from their blog post in march,
We use an adaptive PoW as a rate control mechanism. For honest nodes, the PoW difficulty is relatively low. But if a node begins to spam, the difficulty exponentially increases, physically preventing them from issuing more messages.
Without any official comment, I am concluding that they intend to remove POW or adaptive POW in favor of simply just using Mana as a rate controller. Making it exponentially harder to send transactions in adaptive POW is essentially the same thing as just blocking transactions; it makes sense just to remove it.