Chen's conjecture on positive-density shifted Romanoff type sumsets

Let $\mathbb{N}$ be the set of natural numbers and $\mathbb{P}$ the set of primes. Let $r_1,\ldots,r_t$ be positive numbers with $r_i\geq 1$ for $1\leq i\leq t$ and $r_1^{-1}+\cdots+r_t^{-1}\geq 1$, and define

$$\mathcal{A}=\left\{2^{\left\lfloor m_1^{r_1}\right\rfloor}+\cdots+2^{\left\lfloor m_t^{r_t}\right\rfloor}:m_i\in\mathbb{N},\ 1\leq i\leq t\right\}$$

and

$$\mathcal{R}_{\mathcal{A}}=\mathbb{P}+\mathcal{A}:=\{p+a:p\in\mathbb{P},\ a\in\mathcal{A}\}.$$

Chen's conjecture. If $r_1,\ldots,r_t$ are positive integers with $r_1^{-1}+\cdots+r_t^{-1}\geq 1$, then $\mathcal{R}_{\mathcal{A}}\cap(\mathcal{R}_{\mathcal{A}}+2)$ has positive lower asymptotic density, where

$$\mathcal{R}_{\mathcal{A}}+2=\{n+2:n\in\mathcal{R}_{\mathcal{A}}\}.$$

This conjecture is the subject of a conditional proof under a weak uniform Hardy–Littlewood assumption; the source does not establish it unconditionally.