Minimum diameter conjecture for flip graphs of 3-dimensional polytopes

Let $P$ range over all $3$-dimensional polytopes with $n$ vertices, and let $f$ range over all generic linear functionals on $P$. Write $G(P,f)$ for the flip graph associated with the monotone structure induced by $f$. Minimum diameter conjecture. The minimum diameter of $G(P,f)$ is

$$\left\lfloor \frac{n+5}{4}\right\rfloor$$

for every $n\geq 4$. This minimum can be achieved by simple polytopes for every even $n$. The conjecture concerns the expected lower bound for flip-graph diameters in dimension three; the source presents it as an open problem.