It would seem to me that breaking half shafts would be more likely with an LSD.
With an open diff, the tire with less traction gets all of the torque. So the side with more traction is seeing none.
With an LSD, torque will be transferred to the side with more traction up to the bias ratio, so now the side with more traction is getting more torque than the side with less traction.
Obviously, more torque is more likely to snap an axle.
Lets imagine a scenario where one side sees 100 lbs of static friction force between the tire and pavement while the other side sees 200 lbs. Lets say for the sake of argument that the axle can withstand 150 ft. lbs. of torque. Assume a bias ratio of 2.5:1. Also assume an overall tire diameter of 2 feet.
The maximum amount of torque seen by the axle connected to the wheel with less traction is 100 lbs. * 1 ft. = 100 ft. lbs.
So with a bias ratio of 2.5:1, up to 250 ft. lbs can be sent to the side with more traction. However, the maximum amount of torque this wheel can take without slipping is 200 lbs * 1 foot = 200 ft. lbs.
Now for the relevant part. If this car had an open diff, only 100 ft. lbs. would be seen by both sides, causing one wheel to spin (1:1 bias ratio) and neither axle to break (100 < 150). Because this car has an LSD, 100 ft. lbs. is seen by the axle with less traction, but 200 ft. lbs. is seen by the axle with more traction, causing the axle to break (200 > 150).
Now if the car in question had an open diff, with the same axles, but one tire developed 200 lbs of friction force, and the other 250 lbs, the axle would still break because the torque seen by the axle with less traction, (200 ft. lbs.) is still greater than the torque the axle can handle (150 ft. lbs.).
But wait, you are saying, wouldn't both axles break?
Not likely. We are assuming both axles can take 150 ft. lbs. but in the real world, manufacturing tolerances lead to axles being able to handle slightly different torques. For instance if one axle could take 149.9 ft. lbs. and the other 150.1 ft lbs, the weaker axle would break before the stronger one.