I don’t know if this the right one for this situation, but it’s pretty great. It might be too expensive outside of the US as well
The Audioquest is like 500€ here 
I like the K712pro, but I’ve read that they need an amp otherwise I’ll get poor volume and quality.
I haven’t tried them myself, but I have heard from some people that they are fine on a phone, but also scale fairly well too.
Truth is pretty much everything sounds better with a better amp and/or DAC.
For the most part yes, they do preform better with an amp
You might be able to have a dt770 250 ohm work from the phone
k371 might be better than the 770 250 on a phone… though I believe op has a better phone than me in that regard
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I was actually going to say that, but I didn’t know if it would be right or not
on my phone at least (s10+) I prefer the k371 to the 770 250 by quite a big margin
thought op wanted open back though 
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That’s what gave me pause, cause I was looking for an open back lol
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