I can offer only partial answer to your question, and that is on how this shampoo works.
From The Electronic Medicines Compendium:
Mechanism of action
Coal tars suppress DNA synthesis in hyperplastic skin inhibiting mitotic activity protein synthesis and cell reproduction.
Coal tar decreases epidermal proliferation and dermal infiltration and thus ...
Malassezia overgrowth is thought be the cause of most cases of dandruff (1) and rotating treatments avoids the development of resistance. However, some have argued that overgrowth of Malassezia is a result rather than a cause of dandruff (2).
1 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4852869
2 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2887514
All chemical reactions including killing cancer cells with poisons occur at approximately twice the rate per ten degrees. So reducing the scalp by ten degrees while chemo is occurring will reduce the rate of poisoned hair follicles by half during the procedure.
Currently, there seems to be no convincing evidence to say that alternating between shampoos prolongs their effectiveness in treating dandruff.
1) According to the Scientific Committee on Cosmetic Products and Non-food Products Intended for Consumers by European Commission, 2012:
There is at present no scientific evidence of development of
The study did not investigate the causes of, or possible treatments of, grey hair. However, the research focused on vitiligo, specifically looking at segmental vitiligo (NHS, 2013).
The NHS went further by saying that:
Though the blame for the poor reporting of the study can be put at the door of the press office of the FASEB, which issued a press ...