Completeness of $\{\lfloor 2^k\alpha\rfloor\}\cup\{\lfloor 2^k\beta\rfloor\}$ for irrational $\alpha/\beta$ (Erdős #354)

Let $\alpha,\beta\in\mathbb{R}_{>0}$ with $\alpha/\beta$ irrational, and form the multiset $\{\lfloor\alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 4\alpha\rfloor,\ldots\}\cup\{\lfloor\beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 4\beta\rfloor,\ldots\}$ from the two sequences $\lfloor 2^k\alpha\rfloor$ and $\lfloor 2^k\beta\rfloor$, $k\geq0$. Is this multiset complete, i.e. can every sufficiently large natural number $n$ be represented as $$n=\sum_{s\in S}\lfloor 2^s\alpha\rfloor+\sum_{t\in T}\lfloor 2^t\beta\rfloor$$ for some finite $S,T\subset\mathbb{N}$ (each index used at most once)? What if the base $2$ is replaced by some $\gamma\in(1,2)$?

Acceptance. FULLY RESOLVES: a complete proof that the multiset is complete for all $(\alpha,\beta)$ with $\alpha/\beta$ irrational, or a determination of exactly which $(\alpha,\beta)$ give completeness (settling Hegyvári's conjectured characterization), with all steps. ADVANCES (each independently checkable): (a) prove completeness or incompleteness for an explicit class of $(\alpha,\beta)$ not already covered by the results stated in the background (Hegyvári's dyadic cases, the $\beta=2^k\alpha$ incompleteness of Jiang–Ma / Fang–He, or van Doorn's $\alpha<2<\beta<3$), with proof; or (b) resolve the base-$\gamma\in(1,2)$ variant for a new class; or (c) give a reproducible, rigorously verified representation of every integer in $[N_0,N]$ for a specific previously-open $(\alpha,\beta)$ up to a record $N$, with a certificate. Deliver a proof, or code plus the verified representation range and certificate.