The rank of a word in a deterministic finite automaton is the size of the image of the whole state set under the mapping defined by this word. We study the length of shortest words of minimum rank in several classes of complete deterministic finite automata, namely, strongly connected and Eulerian automata. A conjecture bounding this length is known as the Rank Conjecture, a generalization of the well known Černý Conjecture. We prove upper bounds on the length of shortest words of minimum rank in automata from the mentioned classes, and provide several families of automata with long words of minimum rank. Some results in this direction are also obtained for automata with rank equal to period (the greatest common divisor of lengths of all cycles) and for circular automata.