The aim of this work is to study the capabilities of experiments at the Future Circular Collider FCC-$ee$ to unravel the quark transitions $b\to s\tau^+\tau^-$, unobserved to date. At meson scale the decay $B^0_d\to K^*\tau^+\tau^-$ with the transition $\tau\to\pi\pi\pi\nu_{\tau}$ is studied with a method to reconstruct explicitly the two undetected neutrinos. The detector requirements to study this decay are evaluated, in particular the vertexing resolution performance are emulated and compared to IDEA, a detector concept used for FCC-$ee$ simulations. Analysis of simulated signal events together with simulated dominant backgrounds has been done in order to draw the precision of the $B^0_d\to K^*\tau^+\tau^-$ branching fraction measurement as function of the vertexing resolution performance and evaluate the feasibility of this measurement at FCC-$ee$.