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Recently, the LHCb Collaboration reported pronounced structures in theinvariant mass spectrum of $J/\psi$-pairs produced in proton-proton collisionsat the Large Hadron Collider. In this Letter, we argue that the data can bevery well described within two variants of a coupled-channel approach employing$T$-matrices consistent with unitarity: (i) with just two channels, $J/\psiJ/\psi$ and $\psi(2S)J/\psi$, as long as energy-dependent interactions in thesechannels are allowed, or (ii) with three channels $J/\psi J/\psi$,$\psi(2S)J/\psi$ and $\psi(3770)J/\psi$ with just constant contactinteractions. Both formulations hint at the existence of a near-threshold statein the $J/\psi J/\psi$ system with the quantum numbers $J^{PC}=0^{++}$ or$2^{++}$, which we refer to as $X(6200)$. We suggest experimental tests tocheck the existence of this state and discuss what additional channels need tobe studied experimentally to allow for distinctive tests between the twomechanisms proposed. If the molecular nature of the $X(6200)$, as hinted by thethree-channel approach, is confirmed, many other double-quarkonium statesshould exist driven by the same binding mechanism. In particular, there shouldbe an $\eta_c\eta_c$ molecule with a similar binding energy.

Coupled-channel interpretation of the LHCb double-$J/\psi$ spectrum and hints of a new state near the $J/\psi J/\psi$ threshold