Hankel determinant of incomplete gamma functions

I have some expressions that involve Hankel determinants of incomplete gamma functions. The are of the (rxr form) enter image description here

My problem is that I’d like to evaluate these determinants. Elementary operations help, but these determinants are so ill-conditioned (large n,r) that I have yet to find any technique (equilibration, etc) that provides sufficient stability. There is a body of work (KRATTENTHALER, Advanced Determinant Calculus) that casts the evaluation problem as a continued fraction, but actually working with these expressions is beyond my competence. I am looking for any useful hints.
There is a more general problem of this type involving different functions, but if I can get some help understanding this problem, then I may be able to do the others. Happy to share – this is old work that was never completed.