Adding a module to teach Mathematica the Laplace transform of particular Mittag-Leffler functions

Mathematica 11.3 is not aware of a useful Laplace transform

LaplaceTransform[t^(-a) MittagLefflerE[a, a, t^a], t, s]

which for 0 < a <= 1, Re[s] > 1 should be

(s^a - 1)^(-1)

as easily checked

s = 2; a = 1/2; Print[
 NIntegrate[
  Exp[-s t] t^(-a) MittagLefflerE[a, a, t^a], {t, 0, Infinity}], "=",
 (s^a - 1)^(-1) // N]

(a similar remark was made in Mittag Leffler function Laplace transforms with Mathematica )

I guess it should be possible to add this simple rule via a module. This would allow for example inverting Laplace transforms which are rational functions of s^a, which may be one of the most basic applications of these particular Mittag-Leffler functions.