  In our previous two papers, we constructed an $r$-spin theory in genus zero
for Riemann surfaces with boundary and fully determined the corresponding
intersection numbers, providing an analogue of Witten's $r$-spin conjecture in
genus zero in the open setting. In particular, we proved that the generating
series of open $r$-spin intersection numbers is determined by the genus-zero
part of a special solution of a certain extension of the Gelfand-Dickey
hierarchy, and we conjectured that the whole solution controls the open
$r$-spin intersection numbers in all genera, which do not yet have a geometric
definition. In this paper, we provide geometric and algebraic evidence for the
correctness of this conjecture.
