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At this writing, for Jan. 3 the WSJ site cited in the market description has an S&P 500 open of 3853.29. This is slightly different from the number in the description.

As written ("from open on 3 January") this would say that if SPX closes this week/year at, say, 3900, opens Jan. 3 at 3800, and closes on 31 Jan. at 3850, it would resolve to YES. My natural interpretation of the headline "in January" is relative to December's close. Changing "in" to "during" could help avoid such misinterpretation.

Good point. My consideration was that any change between December's close and January's open is not manifestly attributable to either month. I'd consider it 'increase from December to January', whereas 'increase in January' refers to the change from January's open to close.

Will S&P 500 increase from open on 3 January 2022 to close on 31 January 2022?

I think you mean 2023

@StevenK Yes. You are correct. Thanks.