  In their 2002 paper, Ciucu and Krattenthaler proved several product formulas
for the number of lozenge tilings of various regions obtained from a centrally
symmetric hexagon on the triangular lattice by removing maximal staircase
regions from two non-adjacent corners. For the case when the staircases are
removed from adjacent corners of the hexagon, they presented two conjectural
formulas, whose proofs, as they remarked, seemed at the time "a formidable
task". In this paper we prove those two conjectures. Our proofs proceed by
first generalizing the conjectures, and then proving them by induction, using
Kuo's graphical condensation method.
