A sharp Euler-sum identity for height-one motivic multiple mixed values

Let $d$ be a positive integer with $d\ge 3$. Write $1_{d-3}$ for a string of $d-3$ entries equal to $1$, let $\zeta^\sharp$ denote the Euler sharp sum, and let $\zeta(m,n)$ denote a double zeta value. The sharp Euler-sum conjecture. For all such $d$,

$$\zeta^\sharp(2,1_{d-3},\overline{2})=\displaystyle\!\frac12\zeta(d+1)-2\displaystyle\!\sum_{m+n=d+1,\,2\mid n}\zeta(m,n).$$

This identity is proposed from numerical computation in the discussion of the height-one family $V^\mathfrak{m}_0(d)$.