There are too many independent variables and unknowns for there to be an easy experiment. In any case, I'd be happy for other kinds of evidence. If Twitter is silencing tweets about United, is it part of a corporate ad package, in which case, other folks would've heard about this kind of deal for other companies?
I mean, randomly pruning negative sentiment tweets for a given topic might allow Twitter to tell United, "Hey, we removed 12,000 negative tweets about your brand" (leaving out the fact that there are 12,000,000 negative United tweets total). Or maybe by removing these random tweets, Twitter reduces the overall spread and diversity of this negative sentiment, as many people are content to just retweet what the most popular Twitter users have said about United. But...what would that fix? Twitter already has algorithms in place that make it hard for you to see things either outside of your own filter bubble or otherwise not popular content.
For example, it's well known that many bots are triggered to auto-reply to Trump tweets, as the space devoted to tweet replies are seen as prime advertising space [0]. But I never see those merch-selling bots. The top replies I see to Trump tweets are either from my network, or are replies from popular (usually political) accounts. I have to dig very far to find those random bot replies.
For me, a major argument against Twitter deleting tweets is that it has many other effective ways to suppress content that would be virtually undetectable without wide-scale analysis of their data.
[0] https://www.theatlantic.com/technology/archive/2016/12/weird...
What you and I both know of Twitter's behavior is unlikely to be shared knowledge between us, unless it's been your job to follow the company for awhile in which case I would have to defer to you. But I still expect you to tell me what it is we should both know.
The Twitter I know is happy to shill for political reasons, that puts the possibility of money being given for damage control at corporate as being plausible to me, since there is more money in corporate than politics and Twitter itself is profit-driven with a shit business model. The Twitter you know is not? We could both be wrong. Without clarifying what you mean by "what we know" I wouldn't be so trigger happy with the rhetoric just yet.
They have shut down Twitter accounts for abusing the TOS, and this process is on a case-by-case basis, and can affect well-known users such as Milo Yiannopoullous and Martin Shkreli:
http://www.breitbart.com/milo/2016/07/19/breaking-milo-suspe...
http://www.breitbart.com/tech/2017/01/08/pharma-legend-marti...
Twitter also deletes content in response to valid legal requests and copyright claims, according to their own transparency report:
https://transparency.twitter.com/en.html
We also know that Twitter has applied crowdsourcing/human editing to search query analysis:
http://blog.echen.me/2013/01/08/improving-twitter-search-wit...
I suspect that Trending Topics also has some curation, or at least a manual blacklist, but couldn't find official mention of that.
We also know that Twitter has been sensitive to the tension between being an unfiltered source of information and a platform that amplifies abuse. In January, the company admitted that it "didn't move fast enough" in dealing with abuse complaints:
https://arstechnica.com/information-technology/2017/01/twitt...
In March, they announced new measures for reporting abusive tweets/accounts. Under a heading "Leveraging our technology to reduce abusive content", they describe new ways in which an abusive account can be limited, including suppressing the visibility of tweets: https://blog.twitter.com/2017/our-latest-update-on-safety
So we've seen that Twitter will shut down accounts and tweets based on an interpretation of TOS (i.e. what constitutes "abuse") in ways that not everyone agrees with. At the same time, Twitter knows that its special sauce is unfettered speech, so much so that it was slow to even act on shutting down accounts openly associated with ISIS: https://www.theatlantic.com/international/archive/2016/02/tw...
The real question would be why are they deleting them at all?
Maybe twitter has a team of lawyers sending them lists of Ids to remove, citing DMCA or whatever.
Of course the hypothetical team of lawyers couldn't keep up, and the whole idea is ludicrous, but twitter might comply.
Who knows?