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(I hesitate to post that question here, but I received on answer on FB:) Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks ...

There is a folklore conjecture that $\operatorname{Ext}^2$ vanishes in the category of geometric $p$-adic Galois representations (i.e. representations that are unramified almost everywhere and de Rham ...