Negative, Ghost-Rider
The whole point of having overload protection (even the classic discrete-element heater style) is so a
3Ø motor that's unattended (lets say on a rooftop) that attempts to start on single phase won't fry itself and require a really difficult rigging and replacement job (bucking a 25hp motor across a roof is not fun!)
The heaters should be sized close to the motor's FLA (full load amps). This will almost always be exceeded when a 3phase motor tries to start on 1 phase, then cannot, sits there drawing mega current with a locked rotor until the motor windings short or something else burns up. But, the discrete elements get hot, like right now (!) and trip the mechanically latched contact that's tied to a bimetal strip-type-thingie. The NC contact for the overload goes open and the relay drops out, because the coil can no longer be energized. Motor saved.
Just pointing out that the single-phase issue is present on both modern styles with a little screw adjuster to set the current trip limit AND the classic discrete heater element style where you look in a catalog table and find it needs 3pc of B86 heater to work correctly.
Same issues will also be present should you get a machine which was wired for 480v 3ph and attempt to run it on 240v 3ph, even if the motor's windings have been configured correctly to run on the lower voltage. The current is more or less doubled on the halved voltage and the overload heaters don't like this. One can run for 1-2 seconds until the overload contact opens up.