The notation on pg 3935 in the handbook is quite sloppy, and so is the statement:
"In general, controlling for variables that have a large effect on the outcome can help
reduce standard errors of the estimates and thus the sample size needed. This is a reason
why baseline surveys can greatly reduce sample size requirement when the outcome
variables are persistent."
This statement seems to imply that they're suggesting this regression is correct:
No. The correct regression is:
where each dummy is interacted with the treatment. The ATE is now the sum of the coefficients on the dummies interacted with the treatment.