Stimulated by the recent experimental progress on the ${T}_{\ensuremath{\psi}\ensuremath{\psi}}$ states, the fully charmed tetraquark spectroscopy is systemically investigated by dint of the Godfrey-Isgur relativized diquark model, the modified Godfrey-Isgur relativized diquark model with the color screening effects, and the nonrelativistic diquark model. The theoretical results of the diquark-antidiquark scenarios propose to interpret the ${T}_{\ensuremath{\psi}\ensuremath{\psi}}(6200)$, ${T}_{\ensuremath{\psi}\ensuremath{\psi}}(6600)$, ${T}_{\ensuremath{\psi}\ensuremath{\psi}}(6900)$, and ${T}_{\ensuremath{\psi}\ensuremath{\psi}}(7300)$ structures as the candidates of the $1S$-wave, $1P/2S$-wave, $1D/2P$-wave, and $2D/3P/4S$-wave fully charmed tetraquark states, respectively. On account of the deficiency of sufficient experimental information, e.g., the parities of the newly observed ${T}_{\ensuremath{\psi}\ensuremath{\psi}}$ states, there are uncertainties about the assignments of the ${T}_{\ensuremath{\psi}\ensuremath{\psi}}(6600)$, ${T}_{\ensuremath{\psi}\ensuremath{\psi}}(6900)$, and ${T}_{\ensuremath{\psi}\ensuremath{\psi}}(7300)$ states. It is demonstrated that the further experimental survey on the $cc\overline{c}\overline{c}$ states, implemented by the LHCb, ATLAS, CMS, and other collaborations, ought to be continued in the future.
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