Is there a simple proof of the weight monodromy conjecture in the case of a curve over a mixed characteristic local field?
It seems to me that you can find a proof in the paper of Raynaud: 1-motifs et monodromie géométrique. Astérisque No. 223 (1994), 295–319. I don't know if this is "simple" but it is about curves and their Jacobian varieties. The argument involves the theory of 1-motives and the rigid uniformisation of abelian varieties.