Divisor criterion for the square-root class of Belyi functions

Let $n$ be an odd positive integer, let $X$ be an algebraic curve, and let $f:X\rightarrow\mathbb{P}^1$ be a Belyi function with monodromy of cycle type $(n,311\cdots1,n)$. Let $P$ and $O$ be the locations on $X$ of the ramifications of order $n-1$, and let $T$ be the location of the ramification of order $2$. Let $\operatorname{Sqct}(f)$ denote the square-root class appearing in the claim.

Square-root class conjecture. One has

$$\operatorname{Sqct}(f)=\{(22\cdots2111,322\cdots2,n)\}$$

if and only if

$$(T)\sim\frac{n+1}{2}(P)-\frac{n-1}{2}(O)$$

as divisors on $X$.

The authors say that this was explicitly verified for $g=1$ and $n=5,7,9$, and present it as a conjectural criterion for adapting the square-root cycle-type class to a divisor-theoretic Galois invariant. The general assertion remains unresolved in the source.