Reduction mod $p$ of semi-stable representations of some super-Breuil weights

We determine the mod $p$ reductions of the semi-stable representations $V_{k,\mathcal{L}}$ of weight $k\in [p+5,2p]\cup [2p+6,3p+1]$ and $v_p(\mathcal{L})<1-k\mathrm{/}2$ for primes $p\ge 5$. In particular, this shows that the techniques introduced in [CG24] involving the $p$-adic and mod $p$ local Langlands correspondences can be used to compute the reduction of $V_{k,\mathcal{L}}$ outside the range $k\in [3,p+1]$. Moreover, this shows that the bound on $v_p(\mathcal{L})$ given by Bergdall-Levin-Liu [BLL23] can be improved, at least for weights $k\in [2p+6,3p+1]$.