  Since seminal work of Bowen, it has been known that the specification
property implies various useful properties about a topological dynamical
system, among them uniqueness of the measure of maximal entropy (often referred
to as intrinsic ergodicity). Weakenings of the specification property called
almost weak specification and almost specification have been defined and
profitably applied in various works.
  However, it has been an open question whether either or both of these
properties imply intrinsic ergodicity. We answer this question negatively by
exhibiting examples of subshifts with multiple measures of maximal entropy with
disjoint support which have almost weak specification with any gap function
$f(n) = O(\ln n)$ or almost specification with any mistake function $g(n) \geq
4$. We also show some results in the opposite direction, showing that subshifts
with almost weak specification with gap function $f(n) = o(\ln n)$ or almost
specification with mistake function $g(n) = 1$ cannot have multiple measures of
maximal entropy with disjoint support.
