Create Point based on location of minimum elevation inside a polygon in QGIS3

For QGIS3.0:

Using DEMs, I am drawing polygons around depressions. Inside that polygon, I need to generate not only the minimum elevation value but the location that value was extracted from; this location needs to be a point by the end.

The DEMs in use have near centimeter resolution over features that are a kilometer or so in diameter, therefore generating random points, or sampling all points methods are computationally intense; the PC is capable but future usage of this technique may be on lower-end computers.

Things tried:

Maximum altitude in polygons

How to plot a point on the highest elevation of a raster on a polygon. QGIS

Is there a way to find the minimum Z point feature inside of each polygon in a polygon layer?

The above-mentioned methods were modified from maximum they were written for into minimum, they return a value but not the precise location or require iteration over randomly derived samples of the feature.

I have been performing a similar technique in ArcGIS but trying to commit to QGIS once and for all. This is the last process in my conversion I have not been able to figure out.