  Equistable graphs are graphs admitting positive weights on vertices such that
a subset of vertices is a maximal stable set if and only if it is of total
weight $1$. In $1994$, Mahadev et al.~introduced a subclass of equistable
graphs, called strongly equistable graphs, as graphs such that for every $c \le
1$ and every non-empty subset $T$ of vertices that is not a maximal stable set,
there exist positive vertex weights such that every maximal stable set is of
total weight $1$ and the total weight of $T$ does not equal $c$. Mahadev et al.
conjectured that every equistable graph is strongly equistable. General
partition graphs are the intersection graphs of set systems over a finite
ground set $U$ such that every maximal stable set of the graph corresponds to a
partition of $U$. In $2009$, Orlin proved that every general partition graph is
equistable, and conjectured that the converse holds as well.
  Orlin's conjecture, if true, would imply the conjecture due to Mahadev,
Peled, and Sun. An intermediate conjecture, one that would follow from Orlin's
conjecture and would imply the conjecture by Mahadev, Peled, and Sun, was posed
by Miklavi\v{c} and Milani\v{c} in $2011$, and states that every equistable
graph has a clique intersecting all maximal stable sets. The above conjectures
have been verified for several graph classes. We introduce the notion of
equistarable graphs and based on it construct counterexamples to all three
conjectures within the class of complements of line graphs of triangle-free
graphs.
