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We can suppose that the leading diagonal (NW to SE) is the ascending sequence of primes, and the reverse diagonal (NE to SW) is the ascending consecutive sequence. If we have these the wrong way round then the row we are looking for will be a descending sequence instead of an ascending sequence, so we can just look for either.

This Python 3 program runs in 577ms.

Solution:The ascending row is: 5, 9, 11, 12.There are two ways to construct the square:

The positions of 6 and 14 are interchanged in the two solutions.

In each case the third row is the ascending one, and the sums of the groups are consecutive numbers from 29 to 38.

Yes, I also got two solutions with the four numbers being 5,9,11 and 12 in the third row.

The ascending numbers are 7,8,9 and 10, and the primes are 2,3,11 and 13.