Yes

Even with the multi-dimensional fixed effects model for panel data where T=22 and N=10, checking for stationarity and cointegration remains important. The multi-dimensional fixed effects model excels at controlling for various types of fixed, unchanging differences across entities and time, but it doesn't inherently address whether the individual data series are stable over time or if they share a long-term relationship. Ignoring non-stationarity can lead to misleading regression results, and overlooking cointegration can mean missing important long-run connections between variables. While formal tests might be less powerful with this data size, visual checks, theoretical reasoning, or including time trends could be necessary to ensure the multi-dimensional fixed effects model provides reliable insights.