Keep emergency contacts silent while wearing Apple Watch and both watch & iPhone muted

Hello,

I have an iPhone XS Max w/ iOS 14.7..1 and Apple Watch SE with watchOS 7.6.2.

I have several individuals set as emergency contacts. My understanding is that this setting makes it so these individuals can always contact me, regardless of do not disturb, or if phone is muted.

This is generally the behavior I’m looking for, except, I cannot ever mute the sound, even when I’m wearing my watch and it is muted.

My logic is that, if I’m wearing my watch and both watch and phone are muted, the haptic feedback from the watch is sufficient for getting in touch with me. But my phone or watch ring and ding with every call or text from my emergency contacts.

Is there any way to silence the sounds but still get these notifications while I’m wearing my watch?

Thanks!