Lift-independence problem in the $p$-adic Simpson correspondence for curves

Let $X$ be a proper smooth rigid analytic variety over a complete algebraically closed $p$-adic field $\mathbf C$. Fix a continuation $\operatorname{Exp}$ of $\exp$. Faltings, in the curve case, and Heuer showed that any lifting $\widetilde X$ of $X$ over $B_{\mathrm{dR}}^+/t^2$ induces an equivalen