  At present, the inverse Galois problem over $\mathbb{Q}$ is unsolved for the
Mathieu group $M_{23}$. Here an overview of the current state in realizing
$M_{23}$ as Galois group using the rigidity method and the action of braids is
given. Computing braid orbits for $M_{23}$ revealed new invariants of the
action of braids in addition to Fried's lifting invariant. These invariants can
be used to construct generic braid orbits and more Galois realizations over
$\mathbb{Q}$ for the Mathieu group $M_{24}$, but until now did not lead to
success for realising $M_{23}$ as Galois group over $\mathbb{Q}$. Thus
$M_{23}/\mathbb{Q}$ remains open. Finally, heuristics for searching suitable
class vectors with regard to the realization of groups as Galois groups are
given.
