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The phase diagram of the spin-1 chain with bilinear-biquadratic and next-nearest-neighbor interactions, recently investigated by Pixley, Shashi, and Nevidomskyy [Phys. Rev. B 90, 214426 (2014)], has been revisited in the light of results we have recently obtained on a similar model. Combining extensive density-matrix renormalization-group simulations with conformal-field theory arguments, we confirm the presence of the three phases identified by Pixley et al., a Haldane phase, a next-nearest-neighbor (NNN) Haldane phase, and a dimerized phase, but we come to significantly different conclusions regarding the nature of the phase transitions to the dimerized phase: (i) We provide numerical evidence of a continuous Ising transition between the NNN-Haldane phase and the dimerized phase. (ii) We show that the tricritical end point, where the continuous transition between the Haldane phase and the dimerized phase turns into a first-order transition, is distinct from the triple point where the three phases meet. (iii) Finally, we demonstrate that the tricritical end point is in the same Wess-Zumino-Witten SU(2)2 universality class as the continuous transition line that ends at this point.