The admissible cone metric count conjecture for cyclically branched coverings

Let $X$ be a $d$-fold cyclically branched covering over an $n$-punctured sphere, defined by branching indices $(d_1,\ldots,d_n)$, and let $g$ be the genus of $X$. For an integer $a$, define

$$a(d_1,\ldots,d_n):=(a\cdot d_1\pmod d,\ldots,a\cdot d_n\pmod d).$$

Admissible cone metric count conjecture. There are at least $g$ admissible cone metrics of the form $a(d_1,\ldots,d_n)$.

This claim concerns the number of admissible cone metrics available for constructing a basis of holomorphic $1$-forms on the covering. The surrounding examples show that different admissible metrics can sometimes determine the same branching data modulo $d$; the source does not provide a resolution of the conjecture.