3 problems

Shen–Williams' conjecture. For every $\alpha\geq 1$ and $k\geq 2$, all $\alpha^k$-multiset permutations can be generated by star transpositions; equivalently, $G(\alpha^k)$ has a H…

Hamilton-connectedness conjecture. For every $\alpha\geq 2$, the graph $G(\alpha,\alpha,1)$ is Hamilton-connected.

Hamilton-1-laceability conjecture. For every integer partition $\ba=(a_1,\ldots,a_k)$ with $\Delta(\ba)=0$, the graph $G(\ba)$ is Hamilton-1-laceable, unless $\ba=(2,2)$.