This is actually a mechanical problem, not a staining problem. Microscopic particles of white pigment have flowed down the wall with the rain and lodged into the pores of the brick. Acid will not change their colour nor dislodge them. They have to be removed mechanically. Brushes tend to not get into the small holes. High pressure water washing is recommended to float the particles out, or shove them in out of sight. Soap is not necessary and may even hinder the floating action. Care must be taken that the pressure is not so high as to adversely affect the mortar. One should experiment with the pressure, nozzle size and angle of attack. Then, of course, the entire wall needs to be treated to not have one clean area
. If desired the wall could be soap washed with the high pressure hose after the paint particles have been floated out -- to get off atmospheric grime.
As usual, start at the bottom of the wall and work up. Saturating the surface below the work area will help to prevent the paint particles from re-settling lower down.