Multiple colums in case_when and mutate

Could someone please help me - it is probably quite simple, but I cannot work this out...
My df consists of 3 column of either Low, Medium or High.
I would like a new column summing up the first 3 column:
Low: if obs in column 1-3 are all low
Medium: If 1 or more of the obs in column 1-3 are medium (or less)
High: If at least 1 of the obs in column 1-3 are high.
I have tried this, but the conditions never accepts "High" ...?

test for highs first, then test for mediums, the lows can be the default and probably dont even need their own elaborate tests. The order of what is evaluated for case_when is important, is the takeaway here.

Dear nirgrahamuk
Such a simple answer to so many frustrations ... 1000 thanks🙏🏼
That did the trick - together with only one | and not two ||.
Again - thank you so much.

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