Wichmann's Conjecture on Optimal Sparse Rulers

A sparse ruler is described by a finite sequence $g = (g_1, \ldots, g_n)$ of positive integers called segment lengths or gaps. Its marks are the partial sums $m_0 = 0 < m_1 < \cdots < m_n = L$ where $m_i = g_1 + \cdots + g_i$, and $L = g_1 + \cdots + g_n$ is the length of the ruler. A distance $k \in \{0, 1, \ldots, L\}$ is measurable if $k = m_j - m_i$ for some marks $m_i, m_j$.

A sparse ruler is complete if every distance $k \in \{0, 1, \ldots, L\}$ is measurable. A complete ruler is minimal if no complete ruler of the same length has fewer segments. A complete ruler is maximal if no complete ruler with the same number of segments has greater length. A ruler is optimal if it is both minimal and maximal.

For natural numbers $r$ and $s$, the Wichmann ruler $W(r,s)$ is the sparse ruler with segment-length sequence

where $a^b$ denotes $b$ consecutive segments of length $a$. This ruler has $4r + s + 2$ segments and length $4r(r + s + 2) + 3(s + 1)$.

Prove that if $g$ is an optimal sparse ruler with more than $13$ segments, then there exist natural numbers $r$ and $s$ such that $g$ equals $W(r,s)$ or the reverse of $W(r,s)$.