The answer is cuticle-coating conditioner. Milady's own consultation guidance warns that heavy, cuticle-coating conditioners can weigh down fine hair, leaving it flat or oily — a direct match to the question's wording.
This is part of a broader matching principle the book teaches: product weight should scale to hair texture. The same passage notes that moisturizing shampoos aren't alkaline enough for an oily scalp, and that dry, coarse hair often needs the opposite — a humectant-rich moisturizing conditioner — showing how consultation choices depend on the specific hair type in the chair.
Milady Standard Barbering, 6th ed., Ch. 11 (Treatment of the Hair and Scalp), p. 281: "Heavy, cuticle-coating conditioners can weigh down fine hair, leaving it flat or oily."
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