In Wightman QFT one can prove spin-statistics theorem rigorously. I was wondering, whether a similar result holds in 2D CFT, namely, that $h,\bar{h}\in \mathbb{Z}$ for bosons and $h, \bar{h}\in\frac{1}{2}\mathbb{Z}$ for fermions?

Or put it differently, since we know that for spin $s$ it holds $s\in\mathbb{Z}$ or $s\in\frac{1}{2}\mathbb{Z}$ by spin-statistics theorem, do we also now that the scaling dimension $d$ is also either $d\in\mathbb{Z}$ or $d\in\frac{1}{2}\mathbb{Z}$? Then the result would follow from $d=h+\bar{h}$ and $s= h-\bar{h}$.