On a conjecture on Romanoff type representations

Let $r_1,\ldots,r_t$ be positive integers, and let $\mathcal A$ be the set of integers of the form $2^{k_1^{r_1}}+\cdots+2^{k_t^{r_t}}$, where $k_1,\ldots,k_t$ are positive integers. For any positive integer $n$, let $r(n)$ be the number of representations $n=p+a$ with $p$ an odd prime and $a\in \ma