Determinant of Hilbert-like matrix

Question

It is known that the determinant of the Hilbert matrix with elements $$ H_{tr}=\frac{1}{t+r-1}, \quad t,r=1,\dots,N $$ decreases exponentially to zero. It can be proven that the Hilbert-like matrix with elements $$ H_{tr} = \int_0^\infty d\omega \, e^{-\omega t} e^{-\omega r} = \frac{1}{t+r}, \quad t,r=1,\dots,N $$ has a similar behavior, see e.g. points VI and VII of this paper. I was wondering whether a similar statement could be done for another Hilbert-like matrix, with elements of the form $$ H_{tr} = \int_0^\infty d\omega \, \omega^{2k} e^{-\omega t} e^{-\omega r} = \frac{\partial^k}{\partial r^k} \frac{\partial^k}{\partial t^k} \left( \frac{1}{t+r} \right) = [(2k)!]^N \frac{1}{(t+r)^{2k+1}} , \quad t,r=1,\dots,N \, . $$