That’s correct! You would have two options.
The option you mention is to use the keep probability (1-p) during training and just multiply with it during testing.
This would however add an additional operation during test time.
We could avoid this operation by scaling during training.
So the second option is to scale the units using the reciprocal of the keep probability during training, which yields the same result and is as far as I know the standard implementation for dropout.
As the training is expensive we can just add this scaling, and keep the inference as fast as possible.