Reduction of \( \zeta(1, 3, \bar 3) \) to depth 2

As an alternating MZV, \( \zeta(1,3,\bar 3) \) is depth 3, according to the Data Mine. Using the theory of motives, one can establish the following reduction, to the depth 2 Nielsen polylogarithm \( S_{5,2} \) and classical polylogs at more general arguments. \begin{align*} \zeta(1,3,\bar 3) = & -\frac{7}{2}S_{5,2}\Big(\frac{1}{2}\Big) -\frac{1377}{26}\Li_7\Big({-}\frac{1}{2}\Big) +\frac{17}{468} \Li_7\Big({-}\frac{1}{8}\Big) -\frac{399}{13} \Li_7\Big(\frac{1}{2}\Big) \\ & -\frac{972}{13} \Li_6\Big({-}\frac{1}{2}\Big) \log (2) +\frac{2}{13} \Li_6\Big({-}\frac{1}{8}\Big) \log (2) -\frac{1421}{26} \Li_6\Big(\frac{1}{2}\Big) \log (2) \\ & -\frac{7 \pi ^4 }{192}\zeta (3) -\frac{2 \pi ^2 }{3}\zeta (5) -\frac{7 \pi ^2}{48} \zeta (3) \log ^2(2) +\frac{7}{48} \zeta (3) \log ^4(2) +\frac{121}{16} \zeta (5) \log ^2(2) \\ & +\frac{49}{32} \zeta (3)^2 \log (2) -\frac{3391 \pi ^4 }{112320}\log ^3(2) +\frac{149 \pi ^2}{7488} \log ^5(2) -\frac{2083 }{87360}\log ^7(2) +\frac{101 \pi ^6}{168480} \log (2) \\ & +\text{(primitives)}\,. \end{align*}

In weight 7, the primitives take the form \( \Li_7(e^{2\pi \mathrm{i} p/q}) \), the real part of which is not just a multiple of \( \zeta(7) \). However, numerically it seems in this case \[ \text{(primitives)} = -\frac{154867}{14976} \zeta (7) \,. \]

Questions
How to prove the numerical version of this identity?
 * 1) Will probably need some complicated weight 7 MPL identities
 * 2) Corresponding identities in weight 6 were already fairly complicated