I looked at the segment you mention where Maudlin discusses this. It precisely corresponds to an experimental proposal made by Zeilinger in 1984 (i.e. 40 years ago). The technical difficulty was discussed in the paper.
GENERALIZED AHARONOV-BOHM AND WHEELER-TYPE DELAYED CHOICE EXPERIMENTS WITH NEUTRONS
There are theoretical issues as well. Zeilinger, of course, has since implemented dozens of delayed choice experiments using entangled particle pairs. At no point has there been any indication that any delayed choice permutation might lead to FTL communication. But what about this specific situation, which does not involve entanglement?
In the type Maudlin (and Zeilinger) mentions, the basic idea for FTL communication is: the particle stream is to have the which-slit information determined after the effect has already passed the observation point of that information. Thus the presentation of an interference pattern - or not - would have "jumped ahead" such that the pattern changes faster than c for a portion of the journey to the screen.
The Aharonov-Bohm effect is fairly complex to analyze, at least in the manner necessary to resolve the asked question. Of course, the off-the-cuff answer is that there is no possibility of FTL signaling. Zeilinger has done further work on the Aharonov-Bohm effect (see here, 2011) but apparently has not considered his 1984 proposal worthy of further exploration.
On the other hand, check out this 2007 experiment: A macroscopic test of the Aharonov-Bohm effect. From the paper:
"Given the pivotal importance of Type-I AB-effects, the current experimental situation is surprising. The absence of forces for the magnetic AB-effect has never been experimentally verified, while the electric AB-effect has escaped detection altogether. The former is the topic of investigation in this letter.
... The need for such a test was pointed out more than two decades ago by Zeilinger, but has not been done for Type-I effects."
"In conclusion, we have shown that there is no force acting on an electron passing by a “macroscopic” solenoid of a magnitude that can potentially explain the AB-effect. All force explanations leading to a time delay can be ruled out."
I would not call this a test per se of the specific question you (and Maudlin) are asking. And in fact in some ways this paper echoes Maudlin's call for more experimental research in this area. But I would say: this idea has certainly been considered by a number of researchers, none of whom themselves see any ongoing chance of locating FTL signaling as a possible outcome.
