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Abstract We propose a stronger formulation of the dispersive (or unitarity) bounds à la Boyd-Grinstein-Lebed (BGL), which are commonly applied in analyses of the hadronic form factors for B decays. In our approach, the existing bounds are split into several new bounds, thereby disentangling form factors that are jointly bounded in the common approach. This leads to stronger constraints for these objects, to a significant simplification of our numerical analysis, and to the removal of spurious correlations among the form factors. We apply these novel bounds to B ¯ → K ¯ ∗ $$ \overline{B}\to {\overline{K}}^{\left(\ast \right)} $$ and B ¯ s → ϕ $$ {\overline{B}}_s\to \phi $$ form factors by fitting them to purely theoretical constraints. Using a suitable parametrization, we take into account the form factors’ below-threshold branch cuts arising from on-shell B ¯ s π 0 $$ {\overline{B}}_s{\pi}^0 $$ and B ¯ s π 0 π 0 $$ {\overline{B}}_s{\pi}^0{\pi}^0 $$ states, which so-far have been ignored in the literature. In this way, we eliminate a source of hard-to-quantify systematic uncertainties. We provide machine readable files to obtain the full set of the B ¯ → K ¯ ∗ $$ \overline{B}\to {\overline{K}}^{\left(\ast \right)} $$ and B ¯ s → ϕ $$ {\overline{B}}_s\to \phi $$ form factors in and beyond the entire semileptonic phase space.