Weight-ten basis conjecture for non-alternating Euler sums

Let an non-alternating Euler sum of weight ten mean an Euler sum whose total weight is $10$ and whose indices contain no alternating signs. The constants under consideration are

$$\zeta(10),\quad \zeta^2(5),\quad \zeta(3)\zeta(7),\quad \zeta(2)\zeta(3)\zeta(5),\quad \zeta^2(3)\zeta(4),\quad \zeta(2)S_{2,6},\quad S_{2,8}.$$

Weight-ten basis conjecture. All non-alternating Euler sums of weight ten can be expressed as a rational linear combination of these seven quantities.

The claim proposes a finite spanning set for the weight-ten non-alternating Euler sums studied in the paper. The excerpt reports numerical checks of specific identities but gives no resolution of the general assertion.