Sharp modular-degree conjecture for families of n-gonal curves

Let $V$ be an irreducible variety of dimension $2g+2n-5$, with $4\leq 2n-2<g$, and let $\mathcal{F}\rightarrow V$ be a family of smooth $n$-gonal curves of genus $g$ with maximal variation of moduli. Sharp modular-degree conjecture. If this family has a rational section, then the degree of the modular map $V\rightarrow M_{g,n-gon}$ is a multiple of $\gcd\\{n,2g-2\\}$. Moreover, this number is sharp: there is no other natural number $d$ that is a nontrivial multiple of $\gcd\\{n,2g-2\\}$ such that every family with maximal variation of moduli and a rational section has modular degree divisible by $d$. This is presented as a weaker consequence of the relative Picard-group conjecture and specifies the asserted optimal divisibility of modular degrees.