Existence of nice nef-partitions for smooth well-formed Fano weighted complete intersections

Let $X \subset \mathbb{P}(a_0,\ldots,a_N)$ be a smooth well-formed Fano weighted complete intersection of multidegree $(d_1,\ldots,d_c)$. A nef-partition for $X$ is a splitting

$$\{0,\ldots,N\}=I_0\sqcup\cdots\sqcup I_c,\qquad \sum_{i\in I_j}a_i=d_j,\qquad j=1,\ldots,c.$$

It is nice if there exists an index $i\in I_0$ such that $a_i=1$. Existence of nice nef-partitions. Every smooth well-formed Fano weighted complete intersection admits a nice nef-partition. The existence of such partitions is important for constructing Landau–Ginzburg models and toric degenerations in Mirror Symmetry; the source presents this as a combinatorial conjecture, with no resolution stated.