ARIS characterization by infinite-vertex-strong connectivity

Let $r$ be a positive integer, let $C_{[0,t_1]}$ be a communication sequence, and let the $\mathrm{I\mathcal{V}SC}$ condition be the condition specified by the source. Let ARIS denote the algorithm defined by the stated equations. ARIS connectivity conjecture. For every positive integer $r$, ARIS achieves average-consensus at time $t=t_1(+)$ if and only if $C_{[0,t_1]}$ satisfies the $\mathrm{I\mathcal{V}SC}$ condition. The conjecture claims that this connectivity condition is both necessary and sufficient for ARIS to achieve the stated finite-time consensus property; the supplied text gives no resolution.