Is this MT2 arbor for a drill chuck milled correctly?

Hi folks, I've got a new Drill Chuck with an arbor that has a Jacobs end on one side and a MT2 arbor on the other, and I think the arbor may have been machined incorrectly, but I'm not sure as I'm new to lathes and such. (This is for use in a wood lathe, but please don't hold that against me. )

In particular, I've borrowed a Rikon 70-100 lathe from a friend. It has a MT2 receiver in the tail stock, and currently has a live center on the tail stock. To remove the live center, I just rotate the hand wheel on the tail stock counter clockwise far enough to get the internal bar to push the live center out of the tail stock.

I just purchased a drill chuck with MT2 arbor (PSI Woodworking Products TM32KL Keyless 1/2-Inch Drill Chuck with a 2 MT Mount - Lathe Turning Tools - Amazon.com) to use on this lathe.

When I mount the drill chuck in the tail stock, it sits farther out (from the opening in the tailstock) than the live center. Also, the removal process of rotating the hand wheel counter clockwise far enough to get the internal bar to push doesn't work because the end of the mount doesn't reach the internal mechanism.

My measurements I took on the two arbors are:


The lengths are the same.

From what I've found in online specs for MT2 tapers, the small end should be 0.5720" and the large end 0.7000". I think this means that the arbor I have was milled incorrectly, but is this "within tolerances"? I don't think so, but I'm not sure.

I think I'm going to return the drill chuck and get a new one, but I wanted to see if y'all thought first.

Lastly, Now that I look at it again, I think the Jacobs drill chuck side may not be right either as the drill chuck doesn't look like it's far enough in.

BTW, here's a pic of the drill chuck (this is after giving it a bit of force (a decent hand push from about 1" out and required a couple of good taps with the extraction bar to get it out) to get into the tail stock.



and here's a pic of the live center for reference:



Thanks.

I have some chucks the tailstock will not pop loose but they are without tangs. Does your MT2 taper have a tang on the end of it? If it does have one I would say yes your MT2 taper is ground improperly. Also without knowing, looking at your dimensions and compared to my chucks your new one seats out about .375" from where my chucks, and your live center do. I would say yes there is a problem with the taper size.
But, Since you purchased the arbor labelled as a woodworking tool, woodworkers don't work with tolerances of less than 1/32" usually, maybe it is within tolerance!