Theoretically you might not have to adjust the scope. Realistically it may not. Different ogive shapes, bullet hardness and in-bore acceleration ensures that the barrel harmonics are not the same.

For example, when launched out of the same barrel at the same muzzle velocity, 155gr Lapua Scenars impact 2 moa lower and 1 moa right at 100m than Berger VLD's in both of my target rifles.