Conjecture on the structure of the extremal tree for odd diameter

Let $T^\#$ be the tree defined in the preceding construction, with $Q$ its distinguished path, $B$ its specified subgraph, and $a$ the specified vertex of $T^\#$. For a path $P_k$ in $B$ starting at $a$, let $k$ denote its length.

Structural conjecture. If $d\ge 5$ is odd, then the following hold for $T^\#$:

1. Every path $P_k$ in $B$ starting at $a$ satisfies

$$k\le \frac{d-3}{2}.$$

Equivalently, $T^\#$ has a unique path of length $d$, namely $Q$. 2. If $P_k$ and $P_\ell$ are paths in $B$ starting at $a$, then

$$|k-\ell|\le 1.$$

These properties describe the conjectured structure of the tree $T^\#$ in the odd-diameter case; the source notes that computational evidence suggests them, while the preceding simpler characterization by a caterpillar is false for some small values of $n$ and $d$.