The lower bound for C-loops with associators of prescribed order

Let $C$ be a C-loop, and suppose that $C$ contains an associator of order $n$. An associator is an element of the form $[x,y,z]=(xy\cdot z)^{-1}\cdot(x\cdot yz)$ for elements $x,y,z\in C$.

Lower-bound conjecture. The order of $C$ satisfies

$$|C|\ge 4n$$

if $n$ is odd, and

$$|C|\ge 8n$$

if $n$ is even.

The preceding construction gives nonflexible C-loops attaining these bounds up to the stated parity-dependent values, so the conjecture asserts that the constructions are order-minimal. The source provides evidence but no proof or resolution.