The quadratic lower-bound conjecture for minimal snake diagrams

Let $L(n)$ denote the quantity measuring the size of a minimal snake diagram. Quadratic lower-bound conjecture. For all $n$,

$$L(n)\geq \frac{n^2}{3}+an+b,$$

for some constants $a$ and $b$. This conjecture asserts a quadratic lower bound for the size of minimal snake diagrams, but the supplied text gives no resolution.