Companion points and locally analytic socle conjecture for Steinberg case

In this paper, we will modify the Breuil-Hellmann-Schraen's (more generally, resp.,Breuil-Ding's) local model for the trianguline variety (resp.,Bernstein paraboline variety) to certain semistable (resp., potentially semistable) non-crystalline point with regular Hodge-Tate weights. Then we deduce several local-global compatibility results, including a classicality result, and the existence of expected companion points on the (definite) eigenvariety and locally analytic socle conjecture for certain semistable non-crystalline Galois representations (in other words, the so-called Steinberg case), under some wild hypothesis on on trianguline variety and the usual Taylor-Wiles assumptions.