Ordinal multivariate regression with regularization (Lasso and elastic net)

I'm looking to do a regression of 80 independent variables (continuous values, though many highly correlated) against 750 responses which are ordinal (1,2,3,4). The lassoglm function will perform a linear model regression with regularization for a binomial outcome but not a multinomial one. mnrfit will do the multinomial (ordinal) regression but without the regularization. Does anyone know how I can use regularization with the multinomial regression?

And a related question, does anyone know how to obtain p-values for optimal predictor coefficients with lasso regularization?

Many thanks in advance.