I have a DEM raster with some small and large areas of NoData. I also have a raster defining zones (the zones consists of a rectangular grid). 

I want to create a new raster marking the zones where there exists any nodata values in the DEM raster.

My initial approach was to use the Zonal Statistics tool, using any statistics type (for example Maximum) and uncheck the "Ignore NoData" option. According to what I can read in the documentation, this should give me an output raster with NoData in the entire zones where the input raster contained any NoData pixels. (I could then perform some trivial operations to give me the desired final result).

However, this doesn't work as expected. The output raster from the Zonal Statistics tool have some holes, but there are several places where there are definitely NoData in the input raster and data in the output.

So, are there any other possibilities of achieving what I want? Or have I misunderstood how I should use the Zonal statistics tool?