ag.algebraic geometry - Deligne-Simpson problem in the symmetric group

Question.
Let $C_1,dots,C_k$ be conjugacy classes in the symmetric group $S_n$. (More explicitly,
each $C_i$ is given by a partition of $n$; $C_i$ consists of permutations whose cycles
have the length prescribed by the partition.) Give a necessary and sufficient condition on
$C_i$ that would ensure that there are permutations $sigma_iin C_i$ with

$$prodsigma_i=1.$$

Variant.
Same question, but now $sigma_i$'s are required to be irreducible in the sense that
they have no common invariant proper subsets $Ssubsetlbrace 1,dots,nrbrace$.

I am not certain how hard this question is, and I would appreciate any comments or observations. (I was unable to find references, but perhaps I wasn't looking for the right things.)

This question is inspired by Jonah Sinick's question via the simple

Geometric interpretation.
Consider the Riemann sphere with $k$-punctures $X=mathbb{CP}^1-lbrace x_1,dots,x_krbrace$.
Its fundamental group $pi_1(X)$ is generated by loops $gamma_i$ ($i=1,dots,k$) subject to the relation

$$prodgamma_i=1.$$
Thus, homomorphisms $pi_1(X)to S_n$ describe degree $n$ covers of $X$, and the problem
can be stated as follows: Determine whether there exists a cover of $X$ with prescribed
ramification over each $x_i$. The variant requires in addition the cover to be irreducible.

Background.
The Deligne-Simpson Problem refers to the following question:

Fix conjugacy classes $C_1,dots,C_kinmathrm{GL}(n,mathbb{C})$ (given explicitly by $k$ Jordan forms). What is the necessary and sufficient condition for existence of matrices
$A_iin C_i$ with

$$prod A_i=1$$
(variant: require that $A_i$'s have no common proper invariant subspaces)?

There are quite a few papers on the subject; my favorite is Simpson's paper, which has references to other relevant papers. The problem has a very non-trivial solution (even stating the answer is not easy): first there is a certain descent procedure (Katz's middle convolution algorithm) and then the answer is constructed directly (as far as I understand, there are two answers: Crawley-Boevey's argument with parabolic bundles, and Simpson's construction using non-abelian Hodge theory).

The same geometric interpretation shows that the usual Deligne-Simpson problem is
about finding local systems (variant: irreducible local systems) on $X$ with prescribed local monodromy.

So: any remarks on what happens if we go from $mathrm{GL}(n)$ to $S_n$?

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