“…Second, it is often challenging to relax an assumption involving a finite exponential moment condition to a finite variance condition: in particular, the proof of Theorem 1.2 uses different techniques than in [17,Theorem 13], and new ideas. We emphasize that until now, convergence towards the Brownian CRT of similar rescaled discrete weighted tree-like structures has mostly been obtained under finite exponential moment conditions (see [17,Theorems 1,13 and 14], [48,Theorem 5.1], [53,Theorem 6.60], and in particular the discussion in [51,Section 3.3]). Third, the method is robust, as it allows to treat the case σ 2 µ = ∞, which was left as an open question in [18].…”