Vocab question via math strategy

GRE Verbal Problem Solving
1

I found this question on the skill building support/contrast test and had a tiny doubt.

The question is - Morrison argues that economic prosperity, however widely _________, will not extinguish crime. Many people imagine that all the evils afflicting society spring from want, but this is only partially true.

Now as per me, however is a contrast word so it’s negative (-) and the blank supports the “will not extinguish crime” idea due to the comma (,) so another negative here. giving us a + idea in blank.

Another way of thinking I had was “however” being negtive (-) and economic prosperity being good, the entire idea of blank must be negative (-) hence the blank must be negative.

But the semantic approach I used makes me lean toward the blank being + and the blank being something like distributed, disseminated. Which of the approaches is correct? And am I making any mistakes here?

2

Now as per me, however is a contrast word so it’s negative (-) and the blank supports the “will not extinguish crime” idea due to the comma (,) so another negative here. giving us a + idea in blank.

Firstly, “widely________” and “will not extinguish crime” are two ideas. Secondly, there is a contrasting word in it “however” so, two idea must be going against each other. You guesses are correct or you can add more positive words like " accepted, praise".

The problem with your explanation is part that I have marked in a bold phrase.
Comma does not have significant impact in this sentence because it is overshadow by “however”.