Existence of MDS LCD codes on the maximal hyperelliptic curve

Let $q$ be a prime power with $q\geq 3$, and let $\mathcal{X}$ be the maximal hyperelliptic curve over $\mathbb{F}_{q^2}$ given by

$$y^2=x^q+x.$$

Let $u_B\in\mathbb{F}_{q^2}[x]$ be an irreducible polynomial of degree $q$ satisfying the square condition of Proposition 7 and the MDS criterion of Proposition 6. Let $\mathcal D$ and $G$ be the divisors used in the construction, and let $\mu$ be the multiplier obtained from the square condition. MDS LCD-code existence conjecture. There exists such a polynomial $u_B$ for every $q\geq 3$, and the associated code

$$C=\mu\cdot C_{\mathcal L}(\mathcal D,G)$$

has parameters $[2q,q,q+1]_{q^2}$. The claim is motivated by a counting heuristic comparing the number of forbidden Jacobian classes with the size of the Jacobian; explicit examples were verified for $q=4,5,7$, while existence for every $q\geq 3$ remains open.