Identity for the multiple t-value $t(\{2\}^N,1)$

Let $N\in\mathbb{N}$, and write $t(\{2\}^N,1)$ for the multiple $t$-value with $N$ consecutive entries equal to $2$, followed by $1$. Let $I(2N)$ and $\eta(s)$ denote the quantities used in the source. Identity for $t(\{2\}^N,1)$. For every $N\in\mathbb{N}$,

$$t(\{2\}^N,1)=\dfrac{I(2N)}{(2N)!}=\dfrac{1}{2^{2N}}\left(\sum_{j=0}^{N-1}\dfrac{(-1)^j\pi^{2N-2j}}{(2N-2j)!}\,\eta(2j+1)+(-1)^N2(1-2^{-2N-1})\zeta(2N+1)\right).$$

The source notes that this multiple $t$-value belongs to the basis predicted by Saha's conjecture, but the displayed assertion is presented as an identity rather than as an unresolved conjecture; its status is therefore unclear from the supplied text.