For an understanding of the process that has been happening at Everton, and some insight as to what Meis 'may' have been referring to, if it was even Everton related at all, then this document should help with some insight, or at least some direction for the supposition.
Design and build: detailed design (design by consultant team) - Designing Buildings Wiki - Share your construction industry knowledge. This stage is concerned with developing the detailed design and technical design for the project. It is also likely to include making a detailed planning...
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My thoughts.
We are moving into the build stage. During this stage the Main Contractor effectively becomes the client and with that, the focus changes. During stages 0-3, the lead designer (Meis) is end client (Everton) appointed and they will sit over the design team that will consist of, in our case, Sheppard Robson as local Architects, and Buro Happold as MEP and Structural & Civil consultants (there are more, but these are the key team). Meis will consult very closely with Everton to ensure that the design decisions suit the need and the brief is completed, a monstrous amount of work on a project of this scale.
Once this passes Planning permission it moves into the new stage. Laings have been appointed by the club to realise their vision, and through doing that they give them full autonomy and the ability to create their own team. they have the opportunity to Novate (transfer) the client team or to create their own. Now we know that Laings have employed Pattern Architects as the Technical Architect, and I suspect that Buro Happold have been novated. The end Client cant be unprotected though, so will have a team working on their side to ensure that they aren't screwed over and that any changes are reasonable and within brief. If the original team aren't novated by the Main Contractor, then often they will stay on, but not always, and maybe this could be where any friction lies.
Two things to finish on though. There is no evidence available whatsoever that Meis was referring to anything Everton, but IF he was and he is off the project, so what. The only thing it affects is the continued inflation of his company coffers.