  We define stationary descendent integrals on the moduli space of stable maps
from disks to $(\mathbb{CP}^1,\mathbb{RP}^1)$. We prove a localization formula
for the stationary theory involving contributions from the fixed points and
from all the corner-strata. We use the localization formula to prove a
recursion relation and a closed formula for all genus $0$ disk cover invariants
in the stationary case. For all higher genus invariants, we propose a
conjectural formula.
