Toro 15-38HXL dies after mowing for a while

Thanks in advance.  I have my fathers old Toro 15-38HXL that has been cutting grass for many years with few problems.  Lately, it completely dies while cutting the grass.  It does not seem to do it with the blade dis-engaged.  The seat safety switch seems to work fine.  The PTO switch also works as does the micro switch under the clutch/brake.  The reverse cutting over-ride also seems to work.  But, after a few minutes of cutting grass and while moving forward under not unusual loads, it completely dies.  It usually starts right back up and gives me another few minutes.  
 
I ordered a new (amazon) ignition coil/magneto/armature.  I took the old one out, cleaned and inspected it, but didn't really notice anything.  I also put in a new spark plug.  While I am waiting for the part to arrive, any other ideas?
 
Also, see the picture.  Any idea what that white wire should be connected to?  It has been dangling there for 5 years.  

Sounds as if your loosing spark...
 
You need to check for spark when it's happening. Carry a spare, known good spark plug.
When it dies, plug in and ground the spare plug to the block and check for spark..
 
As it doesn't seem to happen with the blades disengaged I'd be looking at that switch.
 
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You still need to check for spark when it happens...
 
The stator wouldn't cause no spark, the coil can be bad and once hot-fail...
 
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Tks for the update and congrates!