Zariski's multivariable Poincaré-series conjecture

Let $X$ be a smooth projective surface over ${\mathbb C}$, and let $D,D_1,D_2,\ldots,D_l$ be pseudo-effective ${\mathbb Q}$-divisors on $X$. Define

$$M_{X,D,D_1,D_2,\ldots,D_l}(t_1,t_2,\ldots,t_l)=\sum_{m_i\in{\mathbb N}}h^0\left(D+\sum_{i=1}^l m_iD_i\right)t_1^{m_1}t_2^{m_2}\cdots t_l^{m_l}.$$

Zariski's multivariable Poincaré-series conjecture. The series $M_{X,D,D_1,D_2,\ldots,D_l}(t_1,t_2,\ldots,t_l)$ is rational: there exist $f,g\in{\mathbb Z}[t_1,t_2,\ldots,t_l]$ with $g(0,0,\ldots,0)=1$ such that

$$M_{X,D,D_1,D_2,\ldots,D_l}(t_1,t_2,\ldots,t_l)=\frac{f(t_1,t_2,\ldots,t_l)}{g(t_1,t_2,\ldots,t_l)}.$$

The conjecture generalizes the Cutkosky–Srinivas form of Zariski's conjecture from one divisor to several divisors. It is known when $X$ is a Mori dream surface, but the general case remains open; the authors state that they cannot prove it for all pseudo-effective rational divisors.