Caterpillar conjecture for block graphs maximizing mean CIS order

Let $n\geq 5$, and let $G$ be a block graph of order $n$. Suppose that $G$ has maximum mean CIS order among all block graphs of order $n$. Block-graph caterpillar conjecture. Then $G$ is a caterpillar. This strengthens Jamison's Caterpillar Conjecture from trees to block graphs. The claim has been verified for $5\leq n\leq 11$, but the general case remains open.