Flux Field Question

Magnetic flux follows the path of greatest permeability.

Because the plate has a higher permeability than the surrounding air, the flux will actually distort and compact to stay inside of the steel plate along the flux path length.

However, the flux must enter and exit the plate because single ended flux (research Solar magnetic flux shearing) is a very unstable condition. Flux will reposition through any low permeability material to reconnect with other flux of the proper polarity for its B vector.

In reality, any given magnetic value measured at the pole surface, can be followed spatially to map out a 3D toroidal surface, some of which is inside the magnet. Everywhere along that 'surface' the measurement will be the same and the B vector will be the same (purists will note that the B vector ends at the poles and the H vector exists internally). The magnetic strength of that surface is representative of the surface area (which is why the inverse cube rule primarily applies to permanent magnets rather than the inverse square rule).

Your steel plate only provides a better path for the flux, but the area of those toroidal flux surfaces remains the same. So you can squeeze the shape around but you cannot change its net product. For example, imagine that you use some HYMU-80 to provide a short circuit from one pole to the other in the shortest possible path. ALL of the flux could be contained within the 'keeper'. Because the flux surface area is reduced, the Gauss is increased, one is traded for the other. So conversely, if you stretch one of the flux surfaces out for several meters, the Gauss everywhere on that flux surface is greatly reduced compared to that closest to the magnet.

Iron filings do not accurately portray the flux surface because they themselves provide new flux surfaces and are interactive. The result of this is that we get these 'lines of force' that are odd averages of the total interactions. So we change the landscape by looking at it that way.

You can model your experiment with FEMM or Vizimag to see how the flux follows the higher permeability material until the size constraints force it out of the material. But these are only 2D cross section models.