Both Uses

conjecture
in
The Death of Ivan Ilych
(Auto-generated)

His post had been kept open for him, but there had been conjectures that in case of his death Alexeev might receive his appointment, and...
Chpt 1 *

conjectures = opinions based on inconclusive evidence
Having told his wife at dinner-time of Ivan Ilych's death, and of his conjecture that it might be possible to get her brother transferred to their circuit, Peter Ivanovich sacrificed his usual nap, put on his evening clothes and drove to Ivan Ilych's house.†
Chpt 1

Definitions:

(1) (conjecture)
a conclusion or opinion based on inconclusive evidence; or the act of forming of such a conclusion or opinion

A conjecture can be widely believed, but the word is also frequently used to imply that evidence is insufficient to support a belief.
(2) (meaning too rare to warrant focus)