The answer is pustules. Milady's shave guidelines state this directly: do not proceed with the service if the client has a skin infection or pustules, since doing so could spread the infection to other parts of the client's face or to the barber.
Chapped skin is a real, separate concern in the same section, but it's specifically a reason to avoid hot towels, not to cancel the shave outright. A keloid condition and ingrown hairs are legitimate skin conditions covered elsewhere in the chapter, just not the one the book flags as an outright reason to stop before starting.
Milady Standard Barbering, 6th ed., Ch. 13 (Shaving and Facial-Hair Design), p. 339: "The client's skin must be analyzed before beginning the shave. Do not proceed with the service if the client has a skin infection or pustules. Doing so could spread the infection to other parts of the client's face or to you."
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