It is affirmative in current physics but in physics after current physics there are according to us many more constants. In fact these are new constants including cosmological parameters which will change current physics by enriching it with more efficient means of calculations making current physics transcend towards the ultimate physics which will be more refined and autonomous concerning the determination of the values of its constants.
There's nothing enigmatic about the fine structure constant, nor about the other coupling constants, once it is realized that what matters is how such constants depend on the scale-which only makes sense in a quantum theory, since there isn't any scale dependence in the classical equations of motion, that are only sensitive to ratios of the coupling constants, if more than one are present.
And it is the dependence on the scale that allows to find the attractors of the flow equations and, from them, deduce the value of the coupling constants at any scale. In principle this can be achieved using lattice techniques, in order to go beyond perturbation theory; in practice this is a hard problem,
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