Not yet reviewed by Pith; the record is open.

Over the past two decades, the enigma of the deconfined quantum critical point (DQCP) has attracted broad attention across the condensed matter, quantum field theory, and high-energy physics communities, as it is expected to offer a new paradigm in theory, experiment, and numerical simulations that goes beyond the Landau-Ginzburg-Wilson framework of symmetry breaking and phase transitions. However, the nature of DQCP has been controversial. For instance, in the square-lattice spin-1/2 $J$-$Q$ model, believed to realize the DQCP between N\'eel and valence bond solid states, conflicting results, such as first-order versus continuous transition, and critical exponents incompatible with conformal bootstrap bounds, have been reported. The enigma of DQCP is exemplified in its anomalous logarithmic subleading contribution in its entanglement entropy (EE), which was discussed in recent studies. In the current work, we demonstrate that similar anomalous logarithmic behavior persists in a class of models analogous to the DQCP. We systematically study the quantum EE of square-lattice SU($N$) DQCP spin models. Based on large-scale quantum Monte Carlo computation of the EE, we show that for a series of $N$ smaller than a critical value, the anomalous logarithmic behavior always exists in the EE, which implies that the previously determined DQCPs in these models do not belong to conformal fixed points. In contrast, when $N\ge N_c$ with a finite $N_c$ that we evaluate to lie between $7$ and $8$, the DQCPs are consistent with conformal fixed points that can be understood within the Abelian Higgs field theory with $N$ complex components.