Paul Dirac is obviously referring in this quote to the actual physical origin of the FSC and namely regarding the electron and why this value of $\sim 1/137$ and not to any effective theories as mentioned in a previous answer.
In that sense he stands correct even for today's knowledge.
Unless, someone gives analytically or algorithmically a solution to the physical origin of the FSC, meaning the most fundamental explanation of what this ratio actually physically represents in the electron, the source of the electromagnetic phenomena, which mathematically is a proportionality constant and usually refers fundamentally to geometry, the FSC physical origin is still undiscovered.
Many have tried but all have failed. The FSC $α$ seems tightly embedded to whatever intrinsic mechanics and charge flux manifold the electron particle may have. However, since the SM view of the electron is a dimensionless point massive particle there is no such thing according to the SM therefore no room left for exploration and thus a dead end. The SM simply prohibits any intrinsic mechanics for the electron. Therefore, there is nothing more to say about its origin except of deriving it from its measured intrinsic values like charge $-e$ and angular momentum but these are really not a fundamental explanation of its origin:
$$
\alpha=\frac{1}{4 \pi \varepsilon_{0}} \frac{e^{2}}{\hbar c} \approx \frac{1}{137}
$$
Also, notice that at the time Dirac was referring to the FSC "mystery" the SM was not yet established although he was later on declared as one of the creators of the SM today. He was searching all his life to explain the physical origin of this Universal constant but it is ironic that his theories shaped a model that makes it impossible to explain further.
Of course there is always the Beyond the SM branch of research but this also seems to obey until now the basic axioms of the known SM and dares not to deviate theoretically unless something is proved experimentally and any other theoretical research is characterized as heretic.
However, let remind here that one of the biggest discoveries in quantum mechanics by Goudsmit and Uhlenbeck who first theorized that the electron has physical spin angular momentum which lead after to the experimental discovery of the intrinsic quantum spin (i.e. attributed as not classical in the SM) magnetic dipole moment of the electron, their theory was also initially characterized by many as a heretic view of the known at that time quantum theory but they proceeded anyway to publish their theory.
Maybe some bold researcher(s) will indeed come forward in the future as predicted by Dirac to find the origin of this very important for our Universe constant that will unlock as many argue our deeper understanding in physics.
