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Let $(V,c)$ be a finite-dimensional braided vector space of diagonal type. Weshow that the Gelfand Kirillov dimension of the Nichols algebra$\mathfrak{B}(V)$ is finite if and only if the corresponding root system isfinite, that is $\mathfrak{B}(V)$ admits a PBW basis with a finite number ofgenerators.  This had been conjectured in arXiv:1606.02521 and proved for $\dim V=2,3$ inarXiv:1803.08804, arXiv:2106.10143 respectively.

Finite GK-dimensional Nichols algebras of diagonal type and finite root systems