There cannot be dilation difference between the top and the bottom since they both accelerate with each other.
Sounds like a variation of the barn-pole paradox fallacy. So I looked at it a little closer:
The description seems not to state what frame in which the measurements are being taken. On the one hand it seems to use an accelerating frame since the height of the rocket is constant, but on the other hand it seems to use an inertial frame since it talks about Bob's absolute position instead of his fixed position there at the tail. The clocks are presumed to be synced, which isn't true once the rocket changes frames.
Anyway, it seems to describe a brief acceleration, not departing much from the inertial frame in which I've decided the author is using. Bob accelerates after the light is emitted from above and moves a little, meeting the signal after a short time. That is expected even in classic physics.
So I read: "Because the rocket is accelerating, the distance travelled by the second pulse will not be same (as it would be if the rocket were moving with a constant velocity)."
That is correct I suppose from an inertial POV, but the clocks measuring the durations are not inertial any more. That's not a different dilation of one clock vs the other, it is the same situation as the first measurement, but done in a reference frame where the rocket is no longer stationary. Everything changes: h is less, and the two clocks need to be synced in the new frame if they are to be compared again. But all the measurements are being taken using clocks outside the rocket, stationary in the reference frame chosen, and these clocks are no longer in sync with the ones depicted on the ship.
At least that's what I read from it. Maybe I'm interpreting it wrong. I'm no expert in the GR rules, but this seems to be going about it with some reasoning flaws. Relativity says there should be zero difference in the two situation, which differ only in frame at which say the pulse emission is from a stationary Alice. Both pulses are emitted in an identical accelerating environment, so the two test runs are completely equivalent and should yield the exact same elapsed time as measured by Alice and Bob. This seems not to be the case in the article, so the author is making mistakes.