I heard this yesterday in a conference call.

Problem:
The speaker used the wrong adjective.

Explanation:
The speaker was trying to say that he believed that “they” would not be against something.

The primary definition of the adjective “adverse” is antagonistic or unfavorable in effect or purpose, as in “adverse comments”.

It is a common English blunder to confuse the adjective “adverse” with the adjective “averse”, which lacks the letter “d” and means having a strong feeling of antipathy or opposition, as in “averse to kissing in public”.

In other words, “averse” relates to feelings or emotions and is what the speaker should have used.

Solution:
“I don’t think they’ll be averse to …”