The triple multiple $T$-value sum formula

Let $T(a,b,c)$ denote the multiple $T$-value of indices $(a,b,c)$, and let $T(k)$ denote the single $T$-value of weight $k$. For $k\geq 4$, the indices satisfy $a+b+c=k$, with $a,b\geq 1$ and $c\geq 2$. The triple $T$-value sum conjecture.

$$\sum_{a+b+c=k \atop a,b\geq 1,\ c\geq 2}2^b(3^{c-1}-1)T(a,b,c)=\frac{2}{3}(k-1)(k-2)T(k).$$

This is proposed as an analogue of Machide's formula for ordinary multiple zeta values.