20.110 (2022)

Are there residually finite hereditarily just infinite groups that are
$\qquad$ a) amenable but not solvable?
$\qquad$ b) amenable but not elementary amenable?
$\qquad$ c) of intermediate word growth?
$\qquad$ d) of intermediate subgroup growth?
All examples that we know are either linear (hence Tits Alternative applies), or have a quotient with property (T) (hence cannot be amenable).