The minimal mediated sequence cardinality conjecture

For integers $p$ and $q$ with $0<q<p$, let $N(q/p)$ denote the number of elements in a minimal $(0,p)$-mediated sequence containing $q$. A sequence of integers $A=\{0,q_1,\ldots,q_m,p\}$ is $(0,p)$-mediated if every $q_i$ is the average of two distinct elements of $A$. Minimal mediated sequence cardinality conjecture. If $\gcd(p,q)=1$, then

$$N\left(\frac{q}{p}\right)=\left\lceil\log_2(p)\right\rceil+2.$$

This extends the elementary formula $N(1/p)=\lceil\log_2(p)\rceil+2$ to all reduced fractions $q/p$; the source states the claim as a conjecture and gives no resolution.