I’m not at all knowledgeable in this field. So this is just an abstract guess of mine. In the Wikipedia description of Quantum Entanglement it said that

“It thus appears that one particle of an entangled pair “knows” what measurement has been performed on the other, and with what outcome, even though there is no known means for such information to be communicated between the particles, which at the time of measurement may be separated by arbitrarily large distances.”

I like to make guesses and speculation so here goes.

If particle teleportation exists. Then why not teleportation in both directions. Like a swap. With bidirectional teleportation of entangled particles I think they could solve the opposite value problem.

Here two entangled particles A and B change position with each other at a high frequency.

The swapping frequency should be high since this would make target positions easy to find. Just small changes from the previous position at each swap.

So one solution to the value problem could be that in the time A gets a measurement it actually acts on both particles being present. The other particle is teleported away to the other position with the opposite value.

If this does not work. Maybe the high frequency swap can be replaced by a random high frequency teleportation bounce. Here particles bounce on each others position randomly. That is. A goes to B and back and B goes to A and back. This then happens at a high frequency since both particles need to know the others position with a good precision.