Hell, offer a couple of college boys a keg and a couple of pizzas, and they’ll have those posts out quicker than you can imagine. In a probably very inventive and borderline illegal way, too! 
[shameless hijack]One of my uncles belonged to a thrashin’ ring that used a steam tractor a lot like this one. My cousins and I used to go up the the fence separating my uncle’s from the farm ahead in line for threshing on the day they were expected to finish there. When we saw the threshing machine straw blower spout turn around into the stowed position we would run to the corner of the road. When the tractor came along on the way to my uncle’s farm the driver would stop and we rode the tractor down to the farm. Sniff.[shameless hijack]
Any estimate what the actual force being applied might be? A ton? Two?
Maybe you could leave one in the ground, drill a hole crosswise, and rent it out as a mooring point for hot-air balloons. If they’re that hard to remove, you could secure several balloons to the same post.
And hey, when you moor one too many and pull out the post, problem solved. 
What stumps?
The steam tractor, aka traction engine, pulled them OUT, roots and all.
Awesome operation!