Roth's theorem has two universal quantifies, over irrational algebraic numbers $\alpha$ and over real $\epsilon>0$. Of course the theorem asserts in each instance that the inequality $$|\alpha-\frac{p}{q}|<\frac{1}{q^{2+\epsilon}}$$ has only finitely many solutions in integers $(p,q)$.

I seek confirmation (or not) for the following two historical claims:

Prior to Roth's paper...

1) the statement was not known to hold even for a single specific algebraic irrationality of degree $>2$ (but for all $\epsilon$);

2) the statement was not known to hold for single specific $\epsilon$ no matter how large (but for all $\alpha$).

The most famous precursors (Liouville--Thue--Siegel--Dyson) have $\epsilon$ depending on the degree of $\alpha$, so I feel pretty sure about 2), but less confident about 1).