Analytic coefficient conjecture for the Hoffman star basis

Let $a,b\geq 0$, let $C_r=\frac{2^{2r+1}}{2r+1}$, and let $B^{a,b}$ denote the coefficient occurring in the expansion of the real multiple zeta value with star-star regularization. The weight is the corresponding total weight.

Analytic coefficient conjecture. The equalities for the real multiple zeta values are satisfied with

$$B^{a,b}=1-\frac{2}{C_{a+b+1}}\binom{2a+2b+2}{2b+1}.$$

These coefficient identities are the analytic input used to complete the proof of the motivic Hoffman star basis theorem. The source presents this as the remaining conjecture for the coefficients and supplies no resolution in the paper.