The finite-type dimension conjecture for Nichols algebras of Cartan type

Let $(V,c)$ be a braided vector space with $V=\mathbb C x_1\oplus\cdots\oplus\mathbb C x_{\theta}$ and

$$c(x_i\otimes x_j)=q_{ij}x_j\otimes x_i,\qquad 1\leq i,j\leq\theta.$$

Assume that $q_{ii}=-1$ for every $i$ and $q_{ij}\in\{\pm1\}$ for all $i,j$. For $i\ne j$, define $A_{ij}\in\{0,-1\}$ by $q_{ij}q_{ji}=(-1)^{A_{ij}}$, set $A_{ii}=2$, and let $\Phi^+$ denote the positive roots associated with $(A_{ij})$. The finite-type dimension conjecture. With this notation,

$$\dim {{\mathfrak B}}(V)=2^{\lvert\Phi^+\rvert}$$

if $(A_{ij})$ is of finite type, and

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

otherwise. The preceding proposition proves this in the simply laced cases whose connected components are of type $A_m$, while the conjecture also covers the remaining finite-type and infinite-type Cartan diagrams.