Liouville conformal field theory (denoted LCFT) is a 2-dimensional conformal field theory depending on a parameter γ ∈ R and studied since the eighties in theoretical physics. In the case of the theory on the 2-sphere, physicists proposed closed formulae for the n-point correlation functions using symmetries and representation theory, called the DOZZ formula (for n = 3) and the conformal bootstrap (for n > 3). In a recent work, the three last authors introduced with F. David a probabilistic construction of LCFT for γ ∈ (0, 2] and proved the DOZZ formula for this construction. In this sequel work, we give the first mathematical proof that the probabilistic construction of LCFT on the 2-sphere is equivalent to the conformal bootstrap for γ ∈ (0, √ 2). Our proof combines the analysis of a natural semi-group, tools from scattering theory and the use of the Virasoro algebra in the context of the probabilistic approach (the so-called conformal Ward identities).