Is it possible to cover a $70\times70$ torus (or klein bottle/projective plane) with $24$ squares with side length $1,2,3\ldots,24$?

Is it possible to cover a $70\times70$ torus with $24$ squares with side length $1,2,3\ldots,24$? It is known it cannot be done in a $70\times70$ square, which is a shame as the identity $1^2+2^2+3^2+\cdots+24^2=70^2$ is so nice, but perhaps there's still something good to be found in this vein.

As a bonus, attempting this with a $70\times70$ Klein bottle or projective plane would be interesting, too.

edit: The torus case was resolved in the negative in this question, which leaves just the Klein bottle and projective plane identifications left. I think those are still sufficiently interesting questions that I'll wait to see if they can be answered

I ran this through my tiling solver in 2019/2020. It took about 9 million seconds to answer in the negative. I fixed the unit square in place to avoid doing too much extra work. You can fix it anywhere as the toroid is 'featureless' in the sense that there are no corners or edges, every square is equivalent.

I also did a cylinder (Wrap X), Klein bottle (Wrap X, Wrap Y, Flip X), and a projective plane (Wrap X, Wrap Y, Flip X, Flip Y)

I seem to have missed out the Moebius strip (Wrap X, Flip Y). Of course it's not required, being a subset of a Klein bottle. But looking again at the projective plane tiling, I see that I stopped it well before it completed, after 460 hours. It seems it will take quite some time to complete the brute force search.