The author of the paper actually notes that a slightly different series is suspected to converge to $32/\pi^3$. The good news is that that series is an alternating series, so it's guaranteed to converge to something. But, it converges so slowly that, although the second partial sum differs from $32/pi^3$ by about 0.0004328909249, the 5000th partial sum differs by about 0.0004330860787. Those two numbers only start to differ at the 6th digit beyond the decimal point.
Here it is: https://www.emis.de/journals/EM/expmath/volumes/12/12.4/Guil...
The author of the paper actually notes that a slightly different series is suspected to converge to $32/\pi^3$. The good news is that that series is an alternating series, so it's guaranteed to converge to something. But, it converges so slowly that, although the second partial sum differs from $32/pi^3$ by about 0.0004328909249, the 5000th partial sum differs by about 0.0004330860787. Those two numbers only start to differ at the 6th digit beyond the decimal point.