My understanding is that the airfoil type has little to do with it.

However, there is something I forgot to mention. Most airfoils in the range of model reynolds number have laminar seperation bubbles that change size and location on the airfoil with changes in angle of attack. The moment coefficient versus angle of attack curve is "S" shaped with a negative slope on either side of a positive slope. This can be interpreted as the aerodymanic center's location changing position. This in turn causes the neutral point to shift a bit. With a fixed CG set for neutral stability at the most unfavorable angle of attack, the stability goes from neutral to slightly positive over a small range of angles of attack within the useful range of angles of attack. This can frustrate a perfectly neutral setup a little.

BTW, these shifting laminar seperation bubbles can put a kink in the coefficient of lift versus angle of attack curve for many symmetrical airfoils.