What's interesting about this argument is, if its true, it's an example of a simple conjecture that is true but unprovable. But not like Godel sentences, which can be proved under different axioms, or proved to be unprovable. This would be completely unprovable. And not because of some logical paradox, but simply because there is no mathematical reason for it to be false, it's just a coincidence.
There are also probably many conjectures like this. Freeman Dyson constructed one, that no digits of powers of two, reversed, make a power of 5. The reasoning being that the digits grow big quickly, and the base 10 representation is basically random and uncorelared with it.
I think other conjectures about digits of numbers are similar. E.g. Mathematicians have found it impossible to prove that any numbers are normal, or even if pi contains infinite 5's, and many similar properties. These things may only be probabilistically true.