Heng's NMDS code conjecture for the extended trace code

Let $q$ be a prime power, let

$$\mu_{q+1}=\{x\in \mathbb{F}_{q^2}:x^{q+1}=1\},\qquad D=\mu_{q+1}\setminus\{-1\},$$

and let $\operatorname{Tr}^{q^2}_q:\mathbb{F}_{q^2}\to\mathbb{F}_q$ be the trace map, $\operatorname{Tr}^{q^2}_q(x)=x+x^q$. For $b\in\mathbb{F}_{q^2}$ and $c\in\mathbb{F}_q$, define

$$c(b,c)=\bigl((\operatorname{Tr}^{q^2}_q(bx+b)+c)_{x\in D},-\operatorname{Tr}^{q^2}_q(b)\bigr),$$

and set

$$\widetilde{\overline{C_D}}=\{c(b,c):b\in\mathbb{F}_{q^2},\ c\in\mathbb{F}_q\}.$$

Heng's conjecture. If $q>2$, then $\widetilde{\overline{C_D}}$ is a $[q+1,3,q-2]_q$ NMDS code. This conjecture concerns the remaining family of NMDS codes identified by Heng; proving it establishes the asserted parameters and the NMDS property for this extended trace construction.