Issues with mating and nonlinear analysis simulation in pushing needle through folds

I am attempting to simulate a device through the Nonlinear Analysis simulation feature. The functionality of the device can be seen below

The right part is a needle and it is supposed to slide into the sheath on the left and push through the folds at the other end. In the graphic above I "peeled" one of the folds out of the way for easy illustration, but they would just meet together at a point. It works kind of like a straw pushing through a fast food cup lid, in that the needle is pushing the folds out of the way.

So, the sheath is made of 4 parts (a shaft and 3 folds). The folds are essentially welded to the shaft with mates but not to adjacent folds such that they can be forced to open from the inside by the needle.

Now for the Simulation, I am using a nonlinear analysis where the sheath is rubber and the needle is chrome stainless steel. I have fixed just the shaft of the sheath and used reference geometry to move the needle through the sheath. Also, I set a No-Penetration-Contact between the needle and the inside faces of the folds so the needle wouldn't pierce through. Lastly, no external loads.

I run the analysis and it gets to this point (like seen above) where it meets the fold and then just stops instead of pushing through.

Then it gives the error message: "

Solution failure in a Step>1, it could be due to:

The solution may be at a buckling or limit point, i.e.,

displacements grow large under constant forces. If so,

for force control or contact problems, this is the end

of solution (check out the response graphs).

The solver numerical difficulties:

1. Reduce the Singularity-elimination-factor (.5 or 0.)

"

I've changed Singularity-elimination-factor and convergence tolerance for many simulations and nothing changes. I suspect it could be the materials I'm using but I'm not 100% sure. I would really appreciate any help getting through this error and will be super responsive to show my appreciation.

SolidworksSimulation