There is no experiment which can measure the one-way speeds of light. Standard SR adopts the Einstein clock synchronisation procedure, which tacitly assumes that the one-way speeds of light are equal to the two-way speed (i.e., the round-trip speed).
Professor John D. Norton has an excellent website that discusses this topic. Norton is considered an authority on the science of Albert Einstein and the philosophy of science.
As Norton explains, Hans Reichenbach established that you're free to choose another synchronisation convention. For details (including diagrams), please see the Reichenbach's $\epsilon$ section of Norton's site.
In Reichenbach's notation, the standard Einstein synchronisation uses a parameter of $\epsilon=\frac12$, but any values of $\epsilon$ between $0$ and $1$ give valid synchronisation schemes. If you choose any $\epsilon \ne \frac12$ the one-way speed of light in one direction isn't equal to the one-way speed of light in the reverse direction. Nothing is physically different under such a scheme with $\epsilon \ne \frac12$, but the mathematics of the Lorentz transformation becomes messier.
Here's a brief excerpt from Norton's site.
With $\epsilon=\frac12$, light takes 2 units of time to go forward from A to B and 2 units to return from B to A. With the non-standard $\epsilon=\frac14$, things are quite different. Light takes one unit of time to go from A to B. [...] But the light takes three times as long to return from B to A
So while you can do SR using $\epsilon \ne \frac12$, it would not be practical. It's far more sensible to choose $\epsilon = \frac12$, and use the Einstein procedure to synchronise your clocks.
There's no point in asking what's the true value of $\epsilon$. There is no true value. In some ways, it's like asking what's the true velocity of the Earth. Velocity is always relative, a velocity only has meaning relative to some reference frame, and you're free to choose whatever reference frame happens to be convenient. In cosmology, it's often useful to choose the comoving frame of the CMBR (i.e., the frame where the CMBR is isotropic), but that doesn't imply that the CMBR frame is the absolute frame of the cosmos.
Similarly, the $\epsilon = \frac12$ convention is a useful convention, but it doesn't prove anything about the one-way speeds of light.
The one-way speeds cannot be measured independently of a choice of $\epsilon$. So in a sense, those speeds have no physical existence, they're just a mathematical artifact of the $\epsilon$ associated with your synchronisation convention. In contrast, we can measure the round-trip speed of light, and show that it's invariant, and that's the real basis of special relativity.
We can talk about absolute velocities, and one-way speeds of light, but they don't actually correspond to things that are physically observable.
