  Recently the LHCb Collaboration announced the first observation of nontrivial
structures in the double-$J/\psi$ mass spectrum in the mass range 6.2-7.2 GeV,
and a theoretical coupled-channel analysis of these data performed in Phys.
Rev. Lett. 126, 132001 (2021) evidenced the existence of a new state $X(6200)$
close to the double-$J/\psi$ threshold. Although its molecular interpretation
seems the most plausible assumption, the present data do not exclude an
admixture of a compact component in its wave function, for which a
fully-charmed compact tetraquark is the most natural candidate. It is argued in
this work that the QCD string model is compatible with the existence of a
compact $cc\bar{c}\bar{c}$ state bound by QCD forces just below the
double-$J/\psi$ threshold. A nontrivial interplay of the quark dynamics
associated with this compact state and the molecular dynamics provided by soft
gluon exchanges between $J/\psi$ mesons is discussed and the physical $X(6200)$
is argued to be a shallow bound state, in agreement with the results of the
aforementioned coupled-channel analysis of the LHCb data.
