Does each integer $n>1$ have the form $2^k3^l+p_m-p_{m-1}+\ldots+(-1)^{m-1}p_1$?

For each positive integer $ n$ , let $ s_n$ be the alternating sum of the first $ n$ primes given by $ $ s_n:=p_n-p_{n-1}+\ldots+(-1)^{n-1}p_1,$ $ where $ p_k$ denotes the $ k$ th prime. All the numbers $ s_1,s_2,\ldots$ are pairwise distinct. The main term of $ s_n$ as $ n\to\infty$ is not known. It seems that $ \lim_{n\to\infty}s_n/p_n=1/2$ .

Here I ask a novel question involving $ s_n$ on the basis of my computation.

Question. Is it true that $ $ \{2^k3^l+s_m:\ k,l=0,1,\ldots\ \text{and}\ m=1,2,3,\ldots\}=\{2,3,\ldots\}?$ $

I conjecture that the question has a positive answer. Let $ r(n)$ be the number of ways to write $ n$ as $ 2^k3^l+s_m$ with $ k,l\in\{0,1,\ldots\}$ and $ m\in\{1,2,\ldots\}$ . The sequence $ r(1),r(2),\ldots$ is available from http://oeis.org/A308411. For example, $ r(2)=1$ with $ 2=2^03^0+p_2-p_1$ . On May 25, 2019 I verified $ r(n)>0$ for all $ n=2,\ldots,10^6$ . On May 26, 2019 Prof. Qing-Hu Hou extended the verification to $ 2\times 10^7$ on my request. Based on Hou’s program I have verified that $ r(n)>0$ for all $ n=2,\ldots,10^9$ .

PS: By the way, I also note that the set $ $ \{6^k+3^l+s_m:\ k,l=0,1,\ldots\ \text{and}\ m=1,2,\ldots\}$ $ contains $ 3,4,\ldots,10^9$ , and conjecture that this set coincides with $ \{3,4,\ldots\}$ (cf. http://oeis.org/A308403).