In terms of mathematical proof, I don't know (I am not well versed in those).
Empirically, the way to deal with prospective unit roots within your autoregressive model is as follows...
The way you set up your autoregressive variable, it is not unlikely that every single variable within your model have a unit root and are non-stationary (saying essentially the same thing). That is because your variables are not differenced at all. They are all "level" variables. And, if they are embedded within time series, longitudinal data, they will by definition trend somewhat or a lot, and have unit roots.
The first thing to check is if the residuals of such a model are stationary. And, if they are... then your model could be considered adequately specified and resolving the unit root issue due to Cointegration. In other words, you would have a successful Cointegration model suggesting that even though both sides of the equation of your regression have unit roots, there is a stable relationship between your dependent variable and your independent variables.
Now, even if the above is indeed a successful Cointegration model one may argue this model is still somewhat spurious. And, that you would need to make an effort to remove the unit roots. You would do that through first differencing (not yet second differencing). And, that is instead of taking Ys as variables you would take the change in Ys as variables (Y - Yt-1). This would fully detrend your variables. And, you could test them for unit roots. Most likely, the unit roots are gone. Sometimes, unit roots still remain (but to a far lesser degree). That is because the majority of unit root tests are very sensitive if not overly sensitive. However, you could demonstrate that the remaining unit roots are benign and don't need to be resolved further based on the following set of arguments: 1) your variables are fully detrended (most often it does not make sense to go on to second differencing); 2) again test the residuals, and if they are stationary you again have a successful cointegration model; 3) check for the autocorrelation of residuals, including the Durbin Watson score, those are also good diagnostics of the severity of unit roots (serious unit roots have very autocorrelated residuals with DB score much under 1.4 or so).
I suspect your model has modified would demonstrate that either your model has no unit root or really benign one. This is because autoregressive models have very well behaved residuals. With so many autoregressors it is almost impossible for your residuals to be autocorrelated.