Limit theorem for derivative martingale at criticality w.r.t branching Brownian motion

We consider a branching Brownian motion on R in which one particle splits into 1+X children. There exists a critical value λ in the sense that λ is the lowest velocity such that a traveling wave solution to the corresponding Kolmogorov-Petrovskii-Piskunov equation exists. It is also known that the traveling wave solution with velocity λ is closely connected with the rescaled Laplace transform of the limit of the so-called derivative martingale ∂W_t(λ). Thus special interest is put on the property of its limit ∂W(λ). Kyprianou [Kyprianou, A.E., 2004. Traveling wave solutions to the K-P-P equation: alternatives to Simon Harris' probability analysis. Ann. Inst. H. Poincaré 40, 53-72.] proved that, ∂W(λ)>0 if EX(log+X)^2+δ<+∞ for some δ>0 while ∂W(λ)=0 if EX(log+X)^2+δ<+∞. It is conjectured that ∂W(λ) is non-degenerate if and only if EX(log+X)2<+∞. The purpose of this article is to prove this conjecture.