In the paper "On the Sandpile Group of a Graph" by Cori and Rossin one can find a result related to the structure of the sandpile group of $W_n$. Is there a way to provide a set of generators in parametric form in both the cases when $n$ is even/odd?

Update: the identity is known (source is coming), that is the all 2's configuration. Numerical experience suggests that in case of even values of $n$ two configurations generating the group are $(1,2,1,\ldots),(2,1,1,\ldots),$ in case of odd values these are $(2,1,1,\ldots),(2,2,1,\ldots).$ Probably it is also known, however I have not found yet.