Sills–Wang balanced-ordering conjecture for extremal caterpillars

Let $q(x)=x^TA_kx$ be the quadratic form associated with the Wiener index of a caterpillar, and let $b=(b_1,\dots,b_k)$ have multiset of entries

$$\{b_1,b_2,\dots,b_k\}=\{\underbrace{a_s,\dots,a_s}_{m_s},\underbrace{a_{s-1},\dots,a_{s-1}}_{m_{s-1}},\dots,\underbrace{a_1,\dots,a_1}_{m_1}\},$$

where $a_s>a_{s-1}>\dots>a_1$. When $k$ is much larger than $s$, consider vectors whose entries are arranged as

$$x=\{\underbrace{a_s,\dots,a_s}_{l_s},\underbrace{a_{s-1},\dots,a_{s-1}}_{l_{s-1}},\dots,\underbrace{a_1,\dots,a_1}_{m_1},\dots,\underbrace{a_{s-1},\dots,a_{s-1}}_{r_{s-1}},\underbrace{a_s,\dots,a_s}_{r_s}\}.$$

Sills–Wang's balanced-ordering conjecture. The quadratic form $q(x)$ is maximized by this arrangement, where $|l_i-r_i|\le 1$ and $l_i+r_i=m_i$ for $i=2,\dots,s$.

The conjecture concerns the ordering of repeated decremented degrees that maximizes the Wiener index among the associated caterpillars. The paper states that it disproves this conjecture, so its database status is refuted.