Reprojecting a PNG to Robinson projection

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A while ago, I've found a PNG standard map of the world divided to biomes, and I really liked it for it's simplicity. In order to use it in my project (there are no copyright problems), I am trying to reproject it from Plate Caree (the projection it currently seems to have) to Robinson, so I'll be able to represent distances with more accuracy.

Frusratingly, it seems that a lot of the methods I've found rely on my input image having some information regarding it's current projection, which mine obviously lacks, as it is just an image.

Is there some simple way to reproject this image? Thanks in advance.

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