Stability properties of adapted tangent sheaves on Kähler--Einstein log Fano pairs

Let $(X,\mathrm{\Delta })$ be a log Fano pair with standard coefficients endowed with a singular Kähler–Einstein metric. We show that the adapted tangent sheaf ${\mathcal{T}}_{X,\mathrm{\Delta },f}$ and the adapted canonical extension ${\mathcal{E}}_{X,\mathrm{\Delta },f}$ are polystable with respect to $f^{\ast }{c}_1(X,\mathrm{\Delta })$ for any strictly $\mathrm{\Delta }$-adapted morphism $f:Y\to X$.