No, any alarms would be a "Low Altitude Alarm" which is generated based on obstacle clearance and rate of descent. The parameters in the facility should be set so that it will not alarm if the approach is flown as depicted. I suppose if you really pulled the power off rapidly at the FAF and came down like a stone to non-precision minimums you could probably could cause an alarm in the approach facility and they would call you to "Check Altitude". But that would be because of a rapid descent rate which causes the computer to look ahead and then go into alarm.
DC
Agree
assuming no NOTAM's or other special conditions to the contrary, published LOC only minimums are always available on the ILS and could result in a descent below the GS early in the approach. Of course, assuming the aircraft doesn't bust the LOC minimums he will always have terrain clearance becasue they are published minimums and wouldn't be so if they didn't meet TERPS criteria.
In actual IMC, this could be failure of the G/S on the ground, failure of the G/S in the air or an aircraft equipped with VOR only and hence no G/S receiver (as amny old trainers). Remember an ILS receiver is not required for IFR flight, only a VOR/LOC (or TACAN for mil). It's legal to fly IFR/IMC without an ILS, but probably not smart.
Good idea to practice a LOC only approach in VFR sometimes - just in case