Euler sums in terms of generalized polylogarithm values

Let $w$ be a positive integer. An alternating multiple zeta value is a value of the alternating multiple zeta function of weight $w$. For a vector $\vec{s}$, write $w(\vec{s})$ for its weight, and let $\mathop{\rm Li}\nolimits_{\vec{s}}$, $\mathop{\rm Le}\nolimits_{\vec{s}}$ denote the generalized polylogarithms used in the paper. Also write $\zeta(\inbr{1}_w;\sigma_1,\dots,\sigma_w)$ for the alternating multiple zeta function with signs $\sigma_j$. Euler-sums conjecture. Any value of the alternating multiple zeta function of weight $w$ is a rational linear combination of numbers of any one of the following forms: $\mathop{\rm Li}\nolimits_{\vec{s}}(1/2)$, $\mathop{\rm Le}\nolimits_{\vec{s}}(1/2)$, $\mathop{\rm Li}\nolimits_{\vec{s}}(-1)$, $\mathop{\rm Le}\nolimits_{\vec{s}}(-1)$, with $w(\vec{s})=w$, or $\zeta(\inbr{1}_w;-1,\sigma_2,\dots,\sigma_w)$ with arbitrary $\sigma_j=\pm1$ for $j>1$. This conjecture proposes that alternating multiple zeta values can be expressed through any one of several families of generalized polylogarithm values of the same weight.