Feature selection with L1 regularized logistic regression for multinomial classification

Question

I'm trying to do feature selection for my model using L1 regularized logistic regression. I have 4 target classes (low, middle, high, very high), if I set the multi_class parameter of LogisticRegression as multinomial, then use .coef_ it will give me 4 coefficients lists. Is there anyway I can conduct the feature selection with these 4 coefficient lists? Or can I merge my 4 target classes as 2 and do feature selection like a binary classification?

Could anyone please give me some advise on this? Or is there any better way to do the feature selection for multinomial clasification?

Thank you.