What happens if the blocksize stays at 1MB?

With the current impasse that Bitcoin finds itself in, what are the theoretical and pragmatic implications of the blockchain staying at 1MB for the long term future?

Will this simply mean that transactions will take increasingly longer confirmation times? Or will there be a point where the network will simply not be able to cope with the number of transactions (if the transaction rate continues to increase). I have looked online but haven’t been able to find any answers for the blockchain illiterate that explain this.

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