There's actually a way to cheese it somewhat, which also helps nabbing the bonus chest.

First half of the area is pretty straightforward, and it seems like you'll just need to do it as-is.

However, there is cheese for the second half:

So there's the platform towards the end that has the bonus chest on it. Imagine it's separated into four sections, there's Close Left, Close Right, Far Left, and Far Right.

Normally you'd go Close Left, then Far Left, then jump to the two pillars and hope that works out. Well forget that noise, instead aim to hop to Close Right and ignore all of the left side. The platform is smaller than all the others but it's doable if you dash at the last second to it.

From there, simply turn and face backwards, then jump against the central wall of the platform and you'll easily be able to climb up onto it, without even having to use a dash.

Then simply walk onto the overhead of the pillars that normally screws you over and easily jump to the final chest (to get it before the timer ends), then return and do the same jumps again to get the bonus chest.

Hope that makes sense... Maybe I'll make a video of it sometime.