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Coefficientwise total positivity of some matrices defined by linear recurrences

I will present some recent conjectures and results concerning a particular matrix, T(a,c,d,e,f,g), with polynomial entries in six indeterminates that satisfy a three-term linear recurrence, which appears, empirically, to be coefficientwise totally positive (TP). Under certain specialisations the entries of T(a,c,d,e,f,g) count a variety of combinatorial objects with respect to certain statistics, and in some cases we can show that our matrix is TP. This is ongoing joint work with my collaborators (Xi Chen, Alex Dyachenko, Bishal Deb, and Alan Sokal); an extended abstract containing some of our results can be found here