Motivic Linebarger–Zhao conjecture for multiple zeta star values

Let $a_i,c_i\in\mathbb{N}^{\ast}$ with $c_i\neq 2$, and define

$$\gamma_i=c_i-3+2\delta_{c_i},$$

where $\delta_c=\delta_{c=1}$ equals $1$ if $c=1$ and $0$ otherwise. Define

$$B_0=(-1)^{2a_0-\delta_{c_1}}(2a_0+1-\delta_{c_1}),$$ $$B_i=(-1)^{2a_i-\delta_{c_i}-\delta_{c_{i+1}}}(2a_i+3-\delta_{c_i}-\delta_{c_{i+1}}),$$

and

$$B_p=(-1)^{2a_p+1-\delta_{c_p}}(2a_p+2-\delta_{c_p}).$$

Here \text{\boldmath 1}^{\gamma_i} denotes a string of $\gamma_i$ ones. Motivic Linebarger–Zhao conjecture. Each motivic multiple zeta star value satisfies

$$\zeta^{\star,\mathfrak{m}}(\text{\boldmath $2$}^{a_0},c_1,\ldots,c_p,\text{\boldmath $2$}^{a_p})=(-1)^{1+\delta_{c_1}}\zeta^{\sharp,\mathfrak{m}}(B_0,\text{\boldmath $1$}^{\gamma_1},\ldots,\text{\boldmath $1$}^{\gamma_i},B_i,\ldots,B_p).$$

This predicts an expression of every motivic multiple zeta star value as a motivic Euler sharp sum. The section presents it as conjectural and does not establish the identity in full.