When married filing separately, and each claiming 50% of the mortgage interest deduction , should Box 2 of Form 1098 also be split 50/50 between returns?

My spouse and I are married filing separately. We are each itemizing deductions and each claiming half of the mortgage interest paid from our joint account on our mortgage and reflected on Form 1098. When inputting Form 1098, it is clear we should each only enter half the amount for Box 1 (mortgage interest), but should we also each enter half the amount for Box 2 (outstanding mortgage principal)? Our outstanding mortgage principal is $664k, so this exceeds the $375k cap when married filing separately. But shouldn't we be able to claim the full amount of the interest between the two of us (i.e., we can each claim $332k of our shared $664k mortgage and still each be under the $375k limit)?