Magnetic field of a thick conductor

as explained here, the magnetix field generated by a thick conductor is equal to:

Inside the wire: $$B = frac{mucdot I}{2 cdot pi cdot R^2} r$$

It grows linearly with the radius $r$ because increasing r means increasing concatenated current

Outside the wire: $$B = frac{mucdot I}{2 cdot pi cdot r’} $$

That website then states:

Note that the expressions for inside and outside would approach the same value at the surface if the magnetic permeability were the same.

Ok, it’s correct from a math point of view: if we replace $r=r’=R$ in both equations we get the same result.

But now, let’s consider the tangential magnetic field strength interface condition (I have not considered the normal magnetic field interface condition since in this case B is only tangential to the interface between the conductor and vacuum):

$$(vec{H_2} – vec{H_1}) = vec{n_{12}} times vec{J}$$

where $vec{n_{12}}$ is the unit vector orthogonal to the interface surface and $vec{J}$ is the surface current density per unit length [A/m] on the conductor surface.

Now, both of the materials (conductor and vacuum) were both unable to conduct current, then it would be $vec{J}=0$. So, it would be $H_2 = H_1$ and so, if both materials have the same relative permeability, it would be $B_2 = B_1$, where 2 = outside the conductor and 1 = inside the conductor.

But in this case the conductor is able to conduct current, and it does! So, if we cant ignore $vec{J}$, how can the following statemente be true?

Note that the expressions for inside and outside would approach the same value at the surface if the magnetic permeability were the same.