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...
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.
There are no more uses of "conjecture" in the book.
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She dismissed it as mere conjecture.
It is important to distinguish between conjecture and fact.