Conjecture on the construction code maximizing Kemeny's constant for threshold graphs

Let $G$ be a threshold graph of order $n$, represented by a construction code, and let $\mathcal{K}(G)$ denote its Kemeny constant. Maximizing construction-code conjecture. The threshold graph of order $n$ with maximum Kemeny's constant has construction code of the form $0111..1100000...0001$. This conjecture has been confirmed computationally for all threshold graphs of orders up to $n=30$, but the asserted form is not established in general.