It would seem that it would be 6 because 
(5+7)-(2+4) = 12-6 = 6

Let's prove that:

Let the two consecutive odd integers be 2x+1 and 2x+3,
Let the two consecutive even integers be 2y and 2y+2,

where x and y are integers.

Since both odd integers are greater than both even integers,
the smaller odd integer must be greater than the larger 
even integer, so

2x+1 > 2y+2
2x-2y > 1
x-y > 0.5

so the least x-y can be is 1.

It CAN be 1 because in the example mentioned at the top,

(5+7)-(2+4) = 12-6 = 6

5=2*2+1, 7=2*2+3, 2=2*1, 4=2*2, the case when x=2 and y=1,
and thus x-y = 2-1 = 1.

The difference between the sum of those 2 consecutive odd 
integers and the sum of the 2 consecutive even integers is

(2x+1 + 2x+3) - (2y + 2y+2) = (4x+4) - (4y+2) =

4x+4 - 4y-2 = 4x-4y+2 = 4(x-y)+2,

The least x-y can be is 1, and it CAN be 1, so
the least their difference 4(x-y)+2 can be is when x-y=1,
or 4(1)+2 = 6

So we have proved that the least possible difference between 
the sum of 2 consecutive odd integers and the sum of 2 
consecutive even integers when both odd integers are greater 
than both even integers is 6.

Edwin