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Recently I have started to read about network quantum information theory, where network problems are studied under the classical-quantum channel. For example, capacities of the cq-MAC, cq-broadcast or the cq-interference cannels are studied to characterize the maximum achievable rates possible in such communication scenarios. I found Ivan Savov's PhD thesis to be a really interesting document regarding this issues, and very recommendable for anyone interested to start to study network quantum information theory.

However, in chapter 4.3.1 the author states a conjecture called the three sender quantum simultaneous decoder conjecture, where the existence of a POVM decoder for a cq-channel with three senders is conjectured. Such result is very important to proof most of the results of the thesis and it is an important result in general for network quantum information theory. However, at the time it was unproved, and so it remained as a conjecture. I have been researching to see if such cojecture has been proved, but I have been unable to find a general proof for it (in Classical communication over a quantum interference channel it is proved for an special case, but not in a general way).

Consequently, I was wondering if such conjecture has already been proved and so if it has been, I would like to go through such proof. Note that both references I gave are from 2012, so I assume that advances in the issue have been done by this point.

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