Logarithmic passage-time conjecture for the two-point critical edge distribution

Let $T_n(1,2)$ be the passage time between vertices $1$ and $2$ in the critical random graph model, and suppose the edge passage time has distribution

$$t_e\overset{d}{=}p_c\delta_0+(1-p_c)\delta_1.$$

Thus each edge has passage time $0$ with probability $p_c$ and $1$ otherwise. Logarithmic passage-time conjecture.

$$\frac{T_n(1,2)}{\log\log n}\stackrel{\mathrm{P}}{\longrightarrow}\frac{2}{\log(2)}.$$

The source relates this question to work of Baroni and gives divergence of $T_n(1,2)$ in distribution for this two-point law, but the proposed asymptotic constant remains conjectural.