Main conjecture on non-constant families of covers with specified ramification

Let $k$ be an algebraically closed field of characteristic $p>0$, let $d$ be a positive integer, let $S=\{P_1,\ldots,P_n\}$ be a set of points on $\mathbb{P}^1_k$, and let $l_1,\ldots,l_n$ be positive integers. Suppose there exists a degree $d$ cover

$$f:\mathbb{P}^1_k\rightarrow\mathbb{P}^1_k$$

with ramification locus $S$ such that the differential length of $f$ at each $P_i$ is $l_i$. Main conjecture. There exists a non-constant family of degree $d$ covers $\mathbb{P}^1_k\rightarrow\mathbb{P}^1_k$ with ramification locus $S$ and differential length $l_i$ at each $P_i$ if and only if $l_i\geq p$ for at least one $i$. This conjecture characterizes when the specified ramification data admits a non-constant family, building on the definition of families in which the source is fixed while the target varies. Its resolution is not indicated in the supplied text.