The Borwein–Bailey–Girgensohn series conjecture

Let $S_{\mathrm{BBG}}$ denote the Borwein–Bailey–Girgensohn series, let $M=\log 6$, and write $S_{\mathrm{BBG}}=M+R$. For $t>0$, define the exponential integral by

$$\operatorname{Ei}(t)=\gamma+\log t+\sum_{n=1}^{\infty}\frac{t^n}{n\cdot n!},$$

and set $\mathrm{li}(x)=\operatorname{Ei}(\log x)$ for $x>1$.

Main conjecture.

$$S_{\mathrm{BBG}}=\operatorname{Ei}(\log 3)=\mathrm{li}(3)\approx 2.163\,588\,594\,667\,192\ldots$$

Equivalently, $R=\operatorname{Ei}(\log 3)-\log 6\approx 0.3718\ldots$.

The exact identity $M=\log 6$ is unconditional, so the conjectural content concerns the remainder $R$. If true, this would give a notable appearance of the logarithmic integral in a trigonometric series with integer arguments.