Nobby,
Removing the soiling with anything from a mid-range portable should not be a major problem.
Load a 1.5ltr pump-up sprayer with a micro-splitter - and, starting at the top step, pre-spray. Lightly mist the nose with Eco Spotter.
Agitate this with a stiff hand brush - not scrubbing but with very short strokes (about 1-2"). On the nose do this up/down and side/side.
Extract stair with a 4" hand tool, with the stroke passing over the bend of the nose and back. Do not attempt to scrub with the tool - use fairly slow passes, because what id needed is a good flush-action. Set your solution pressure to 300psi or the best your machine can provide.
Repeat on subsequent steps.
An effective clean could be achieved with detergent-based chemicals but the main active ingredients, being contained in the pre-spray would largely be removed on the first pass. This means that several re-treatments may be required in extreme cases. With the micro-splitter approach, once the soil has been shattered (split), the product has fulfilled its function on a permanent basis, so extended rinsing without re-application happen at a level efficiency.
The above should take care of soiling, however carpets such as this are particularly prone to apparent change of shade due to abrasion and pile distortion. Dogs often drag the front face of their legs against the nose of stairs whilst ascending, whilst human feet slide over the nose. Given that the oil deposits from the dogs both hold on to soil and increase the frictional coefficient of the pile, then the effects of human foot traffic are exaggerated. Consequently, even with all the soiling removed, the nose areas of each stain may appear a different colour to the remainder of the carpet.
Full customer satisfaction cannot be guaranteed.