Knowing that, the question "but is his statement true?" seems to be unneeded essentialism.

My impression is that this is what the whole debate is about. What matters to Corresponders is whether or not statements describe reality. Bob's statement correctly predicts whether or not he has cash on him- and "correct prediction" is their standard for truth. They wouldn't care that his dishonesty and misinformation cancel each other out, but would agree that in general dishonesty and misinformation lead to less correct predictions.