electromagnetism - Classical explanation for volume conservation in magnetorestriction?

This is a follow-up to this question, on a more specific theoretical point. Magnetostriction is the phenomenon discovered by James Joule in which magnetic substances experiences stress when exposed to a magnetic field. Joule says:

About the close of the year 1841, Mr. F.D. Arstall, an ingenious machinist of this town, suggested to me a new form of electro-magnetic engine. He was of the opinion that a bar of iron experienced an increase of bulk by receiving the magnetic condition.

Joule then describes an experiment in which he detected an increase in length of an iron bar in the direction parallel to the field, and then a follow-up in which he immersed the bar in water and tested for a volume change using a capillary tube. He found no volume change, implying that each of the transverse dimensions changed by $-1/2$ the fractional change seen in the longitudinal direction.

The following graph (Great Soviet Encyclopedia, 1979) seems to show that the volume change is only zero for small fields. The slopes of the two curves seem differ by a factor of about $-1/2$ up to a field of about $1.5\times10^3$ oersteds. Assuming a permeability of about $10^4$, this is $B\sim10^7$ gauss, which I think is of the same order of magnitude as the nominal saturation field.

It seems to be a big deal or possibly controversial to see a volume change. (Chopra and Wuttig, Nature 521, 340–343 (2015)), and it only seems to happen in these very strong fields.

It seems clear that according to classical E&M we should expect a longitudinal compression and a transverse expansion, since the stress-energy tensor $T^\mu{}_\nu$ of a uniform field in the $x$ direction is proportional to $\operatorname{diag}(1,1,-1,-1)$ ($+---$ metric, $txyz$ coordinates). However, I would think that the volume would change, since the 3x3 stress tensor looks like $\operatorname{diag}(1,-1,-1)$ (as required by $T^\nu{}_\nu=0$), not $\operatorname{diag}(1,-1/2,-1/2)$ (which would give zero trace for the 3x3 part, but not for the 4x4 tensor, which is what's required).

So is there any classical explanation for the lack of volume change that is usually observed at small fields?