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I think I'll answer this in two separate answers, so as not to conflate in-game issues with real-world issues.

To address the issue in the game world,. it's not quite as easy as issuing a statement of support. After all, It's already known that the church supports Konsi, a lot of people won't see the statement, and nothing they could say would come close to measuring up to open recognition of her deeds by the Open Lord...

The issue isn't really about Konsi's character at all, the issue is that there's an opportunity for opponents of the church to run a smear campaign, and there's no shortage of people who might want to do that.

Agents of Beshaba will not care if the majority of people believe them, and won't be trying to convince everyone. They just want to publicly allege corruption, throw in some doubt and confusion. They want to give ammunition to those who want to be convinced that the church is bad to back up those views. Give them ammunition to reinforce their existing beliefs.

There are plenty of people who dislike how much power the Tower of Luck has, especially over the gambling industry in Waterdeep. Some people don't want to pay fees or employ members of the tower to oversee games, it's almost like guild dues. Some people want to run crooked games, but find it difficult to find marks when all the reputable games are sponsored by the church. Some people believe that it's not the place of the church to be doing any of this, the size and power of the Tower of Luck isn't at all consistent with how the faith is normally represented - there are theologians who argue that Waterdhavian Tymoran faith is a heretical sect.

Some of the nobles who lost in the tournament might have a bit of a bruised ego from "losing to a goblin" - if they can cause a controversy about the match, claim there was a fix they can convince people around them, and themselves that they didn't "really" lose.

And... of course, those people who just absolutely hate, hate, hate goblins (of which there are many) will be inclined to believe the worst, no matter what anyone says or does.

If such attacks were made, would it seriously damage the church? Probably not - but it'll cost them a few people here and there. Worse, it could form the foundation of a sustained assault on the position and power they hold in Waterdeep, and it's the job of the church's leaders to protect that.

I imagine, if anything does happen, there will be more specific details in future comics...

I think I'll answer this in two separate answers, so as not to conflate in-game issues with real-world issues.

Let's talk a little about how real-world conspiracy theories work, and the dangers therein. I don't want to get too into this, because I don't think this is the right forum for such things, but there's some important things to consider.

1. Nobody is a fully rational actor all the time. Regardless of how "easy" something is to debunk there's always going to be a few people here and there who hear it from a source they trust and believe it. It's incredibly wearing to have to independently check every single thing you hear to see if it's true.

2. It's a widely known problem that when presented with a narrative about something they have no experience of, most humans will subscribe to the first viewpoint they're exposed to, and it's significantly harder to convince a person that they're wrong once they've formed an opinion - reason and logic be damned. Plenty of otherwise-rational people might subscribe to something that makes no sense if they've heard it a few times without any pushback.

3. The idea that if it's "only a small number of people" it's not dangerous is a little flawed. Some conspiracies are silly and mostly harmless, and don't put other people in danger. However, conspiracies about a specific person or place can lead its subscribers to "take matters into their own hands", endangering those people or places. We've had plenty of examples of this that have cost lives.

4. A lot of the time, it's less important how many people are subscribed to an irrational or easily debunked crackpot theory, but who those people are. We've seen examples in most developed nations of legislators up to the highest levels promoting baseless and dangerous rhetoric that influences a large number of people to action. Certain celebrities on social media are known for directly making baseless conspiratorial attacks on people, that drive their fans and supporters to take up those attacks as a cause.

I'm refraining from specific examples, because I don't want this to degenerate into an argument about specific examples, however I hope that this makes sense without the need for such.

That, though, raises another question, because if the conspiracy theories are so easy to debunk that no rational person believes them, why worry about their existence?

Most people aren't rational, that's why.