In fact the proof of Proposition 4.9 shows that if we just have that $\tilde{\omega}_{t}\to f^{*}\omega$ in $C^{0}_{loc}(M\backslash S)$ (or in the $C^{2}$ topology of Kähler potentials) then (1.3) holds in the $C^{1,\alpha}$ topology of Kähler potentials. It seems that just having $\tilde{\omega}_{t}\to f^{*}\omega$ in the $C^{1,\alpha}$ topology of Kähler potentials (which is proved in [38] in general) is not quite enough to deduce (1.3).