Does the isothermal compressibility diverge at the lambda line in Helium-4?

Helium-4 has a normal-superfluid phase transition which corresponds to the lambda line in the P-T diagram. The heat capacity at constant pressure $C_p$ shows a discontinuity when the lambda line is crossed. From universal scaling, the susceptibility associated with the order parameter has a divergence at the critical temperature, which can be any of the lambda line. In this case the susceptibility is not a physical quantity.

I read here it is known that the isothermal compressibility diverges at the lambda line but I don’t see directly why. Does it have a critical exponent? Looking at the tables here, like in page 437, I thought the isothermal compressibility doesn’t diverge. If the heat capacity at constant volume is $C_v neq C_p$, by the relation between them,
$$
C_p-C_V= -T alpha^2 VT /kappa,
$$
where $alpha$ is the coefficient of thermal expansion and $kappa$ the isothermal compressibility, then $kappa$ doesn’t diverge. Unless $alpha$ diverges in such a way to make $alpha^2 / kappa < infty$.

I would appreciate if you could give me some references. Thank you in advance.