Another thought occurred to me about this. It seems that wire diameter is important for two reasons.
First, larger diameter helps make the higher resistance of the iron (as compared to copper) less of an issue in creating differences. Any test for inductance needs to be at a frequency well below the R/L frequency of the coil. In other words, when you charge the coil with a square wave voltage signal, you would like to see the current rise linearly, and not exponentially.
Also, greater diameter means more iron, which might make the effect (again, if there is an effect) more noticeable. More iron could cause more distortion of the fringe field.