  We show that Monotone 3-Sat remains NP-complete if (i) each clause contains
exactly three distinct variables, (ii) each clause is unique, i.e., there are
no duplicates of the same clause, and (iii), amongst the clauses, each variable
appears unnegated exactly twice and negated exactly twice. Darmann and D\"ocker
[6] recently showed that this variant of Monotone 3-Sat is either trivial or
NP-complete. In the first part of the paper, we construct an unsatisfiable
instance which answers one of their open questions (Challenge 1) and places the
problem in the latter category.
  Then, we adapt gadgets used in the construction to (1) sketch two reductions
that establish NP-completeness in a more direct way, and (2), to show that
$\forall\exists$ 3-SAT remains $\Pi_2^P$-complete for quantified Boolean
formulas with the following properties: (a) each clause is monotone (i.e., no
clause contains an unnegated and a negated variable) and contains exactly three
distinct variables, (b) each universal variable appears exactly once unnegated
and exactly once negated, (c) each existential variable appears exactly twice
unnegated and exactly twice negated, and (d) the number of universal and
existential variables is equal. Furthermore, we show that the variant where (b)
is replaced with (b') each universal variable appears exactly twice unnegated
and exactly twice negated, and where (a), (c) and (d) are unchanged, is
$\Pi_2^P$-complete as well. Thereby, we improve upon two recent results by
D\"ocker et al. [8] that establish $\Pi_2^P$-completeness of these variants in
the non-monotone setting.
  We also discuss a special case of Monotone 3-Sat-(2,2) that corresponds to a
variant of Not-All-Equal Sat, and we show that all such instances are
satisfiable.
