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To be clear, while I respect that they've taken measures that are more than just empty words - especially considering their poverty compared to the rich Gulf states who are silent or complicit - I'm not putting it past their aims and objectives, especially given their history in Yemen and source of support/funding. But it's worth remembering though that the Houthis stated that their aim was (at least until the US-led coalition stepped in) solidarity and to put pressure on so that aid can enter Gaza.
The US led coalition deserve more blame for the Red Sea escalation. To my statement earlier, they sat back and unconditionally defended Israel and done nothing of worth to help de-escalate the situation, but were quick to go war mongering in the Red Sea, waters where their presence has always been contentious and hated by pretty much all the local populations.
You're right they haven't but that is their aim.