Great question, especially given that we're in the peak of the solar cycle.

It's definitely possible that the 11-year solar cycle has some relation to the strength of the stratospheric polar vortex.  But it's tricky to quantify that relationship because there are other potential sources of variation such as decadal SST variability and the Quasi-biennial oscillation.

Another issue is the length data that we have. If the cycle is 11 years and we only have ~60 years of data, that's about 5-6 independent samples which is not enough data to really understand that relationship. There's been some research looking at it in models, but not many models simulate solar cycle variability that well. This recent paper summarizes some of the issues with simulating the solar cycle and its impacts on the stratosphere in one model: https://acp.copernicus.org/articles/19/5209/2019/