I was one of those people who mentioned back room deals. And it's not that I'm saying that's how it works, but my thinking has nothing to do with price - I agree with you that the index funds are price agnostic - but more to do with availability of shares. In other words they could make a deal to buy a certain number of shares at whatever the closing price is on Friday (for example) but not to buy them on the open market but instead from a pre defined "partner in crime"? This to ensure shares available to them and not get caught up in possible open market chaos - again they are price agnostic - but much harder to match the index well if the price swings wildly in a trading day and in AH trading?
In this scenario, the index fund will be trying to protect itself against the risk that at Friday's Closing Cross (which is the moment when they can buy at exactly the price that will keep them in line with the index) there will not actually be any shares to buy (because also at the closing cross a trade involves not only a buyer and but also a seller, so the volume is limited by the number of willing sellers).
It has already been established that the actual number of shares that a given index fund needs to buy is not that sensitive to the closing price, the risk is that this number of shares will be unavailable at the right time - and thus have to bought later at a potentially higher price.
So assume that to mitigate this risk the index fund has ahead of this event approached someone with this offer: "At the Closing Cross on Friday Dec. 18 we will buy this specific number - X - of Tesla shares from you at "market" price (i.e. the price at the Closing Cross)."
While very different in the details, this contract has some similarity to the index fund buying call options. For example, the further away in time they entered into the contract the more time value it would have due to (even) greater uncertainty of the stock price at expiration. While the buyer of the contract (the index fund) would reduce their risk, the seller of the contract would conversely increase their risk - specifically that at expiry the stock price would be lower than what they would need to deem the contract desirable.
The point is that this deal would only be acceptable to a (rational) supplier of shares for an additional payment, basically a premium to compensate them for the risk they take - similar to what a writer of a call option would expect.
And herein lies the rub: The index fund is supposed to acquire the shares at market price at the Closing Cross on Friday - but trying to do so via some pre-arranged contract will necessarily incur an extra cost to them - causing them to surely deviate from the Index, which was the very thing they were trying to avoid.
So I can only see such a pre-arranged transfer of shares to an index fund from someone who can accept to not get compensated for the risk that deal involves - maybe a situation where an index fund is owned by someone who also owns a different fund that could buy the shares ahead of time and could somehow accept to potentially lose out on the deal.