The 10 minutes is the average time. There are enough cases in which no block has fallen for almost an hour and other cases in which several blocks have fallen within a minute. So there is no guarantee that you will send the transaction to the net just before 10 minutes have elapsed and the next block will fall shortly afterwards. That would be a huge amount of luck that you would also need. The only way out of this dilemma, as already mentioned several times, is to find a large miner who will include your transaction in the next block to be calculated. This large miner must have so much power that he regularly mines blocks and the probability is in your favor that he will ever mine a block. And of course there must be an incentive for this miner, perhaps in the form of a reasonable commission, to comply with your request and make the extra effort to do so. Very unlikely, but not impossible.
All other dubious and questionable methods mentioned here are not target-oriented but misleading and inspire false confidence, just like the last hopeless appeal of a desperate user. Even if you were to find out the key and provide some form of proof here in the forum that you know the key, it would not change the fact that you have no legal right to open the vault and obtain the contents. To make this clear, let's take a simple example: imagine that Alice has known the key from #66 for over a year. This fact alone does not make her the owner of the contents of the safe from puzzle #66. If Mallory finds out the key 3 years later and grabs the coins from #66, they belong to Mallory. That's the way the game is, that's how it's defined in Bitcoin and there is no "law" or "rules". The mighty power is written in Bitcoin protocol and everything is defined in there.