A real matrix $M$ is called totally positive if all of its minors are positive; these matrices have been extensively studied, and there are generalizations to other Lie types, for example by Lusztig.

I am interested in the following $q$-analog: a matrix $M$ with entries in $\mathbb{R}[q]$ is $q$-totally positive if all of its minors are polynomials in $q$ with positive coefficients. I see that one other question asked about this notion in the case of the $q$-Pascal matrix.

Has this class of matrices been studied (other than showing that particular matrices have this property)? Is this the "right" $q$-analog of total positivity? Is there a good Lie-theoretic definition of this class of matrices that works in other types?