Diophantine approximation by primes and Landau--Siegel zeros

Let $\alpha >0$ be an irrational number of finite type and let $\beta \in \mathbb{R}$. Assuming the infinitude of Siegel zeros (respectively, sufficiently strong Siegel zeros), we show that for every sufficiently small $\epsilon >0$, there exist infinitely many primes $p$ such that \begin{gather*} |αp+β|\leq p^{-1/3+\varepsilon} \qquad$\text{(respectively, }$|αp+β|\leq p^{-1/3-1/20}$\text{)}$. \end{gather*}