The $D_4$ and affine $D_4$ dimension conjecture for Nichols algebras

Let $(V,c)$ and the matrix $(A_{ij})$ be as above: $V=\mathbb C x_1\oplus\cdots\oplus\mathbb C x_{\theta}$, $c(x_i\otimes x_j)=q_{ij}x_j\otimes x_i$, $q_{ii}=-1$, $q_{ij}\in\{\pm1\}$, $A_{ii}=2$, and $q_{ij}q_{ji}=(-1)^{A_{ij}}$ for $i\ne j$. Let $\mathfrak B(V)$ be the Nichols algebra of $V$. The $D_4$ dimension conjecture.

$$\dim \mathfrak B(V)=2^{12}$$

if $(A_{ij})$ is of type $D_4$, whereas

$$\dim \mathfrak B(V)=\infty$$

if $(A_{ij})$ is of type $D_4^{(1)}$. This is the weaker special case that the authors say they will need from the preceding finite-type dimension conjecture; its resolution is not supplied in the given text.