I pulled out my Ulmia wood plow plane and began to cut a groove with my 1/8 inch iron. Unfortunately my narrow blade started to bind in the groove as it got deeper.
When cutting that deep you need to make sure that the iron retains the original bevel on the sides and the skates are in line with that bevel or it will get hard to push the deeper you go.
Along with making sure there is a good edge on the side bevel you might try waxing the skate.
One problem with plow planes is as the cut gets deeper if there isn't a bit of clearance any slight misalignment of the plane in the groove can cause the skate and blade to wedge a little. Sometimes with narrow blades it helps, after a bit of depth is achieved, to remove the fence.
jtk
"A pessimist sees the difficulty in every opportunity; an optimist sees the opportunity in every difficulty."
- Sir Winston Churchill (1874-1965)