Let $S$ be a closed oriented surface of genus $\geq 2$ and $\mathcal{F}_n(S)$ be the space of flat metrics with conic singularities on $S$ whose cone angles are of the form $2k\pi/n$ ($k\in\mathbb{N}$). Let $\mathcal{F}_n^1(S)$ denote metics $g\in\mathcal{F}_n(S)$ of total area $1$.

Each $g\in\mathcal{F}_n(S)$ yields a conformal structure on $S$ and hence a unique hyperbolic metric $g_\mathrm{hyp}$ in the conformal class, which can be written as $$ g_\mathrm{hyp}=\lambda_g\cdot g $$ outside the singularities of $g$.

I need some information on the relationship between $g$ and $g_\mathrm{hyp}$, for example,

Question:Fix $n\in\mathbb{N}$. When $g$ runs over $\mathcal{F}_n^1(S)$, does $\int_S\lambda_g^{-1}\mathsf{dvol}_g$ have a upper bound or a positive lower bound?

It seems that when $n=2$, a lower bound exists but upper bounds do not. see the remark below.

It is also natural to ask whether $\max_S(\lambda_g^{-1})$ has upper or lower bounds.

**Remark:**
A metric $g$ belongs to $\mathcal{F}_n(S)$ if and only if it has local expression
$$
g=|a(z)|^\frac{2}{n}|dz|^2
$$
for some conformal structure on $S$ (of which $z$ is a conformal local coordinate) and some holomorphic $n$-differential $\alpha=a(z)dz^n$.

Now assume $n=2$, then the integral in the above question is the Weil-Petersson co-norm of the quadratic differential $\alpha$ (viewed as a cotangent vector to the Teichmüller space). On the other hand, the total area of $g$ is the Finsler co-norm of $\alpha$ with respect to the Teichmüller metric, therefore, for $n=2$, the existence of a lower bound in the above question is equivalent to the majorization of the Weil-Petersson metric by a constant multiple of the Teichmüller metric. The latter majorization is established in a 1974 paper by Linch, but her method is based on Teichmüller geodesics and does not seem to be generalizable to other $n$'s.

Also, I have read that the problem of understanding the relationship between $g$ and $g_\mathrm{hyp}$ is treated by many authors such as Masur, Minsky, Rafi, etc., however, in their papers I only find miscellaneous technical statements on this and can't get a guiding idea. I would appreciate any systematic explanation of that relationship.