The generalized vector space partition conjecture for $A_q(2k,2k-2;k)$

Let $A_q(n,d;k)$ denote the maximum size of a constant-dimension $k$-subspace code in $\mathbb{F}_q^n$ with minimum subspace distance at least $d$. Generalized vector space partition conjecture. For each $k\ge 4$,

$$A_q(2k,2k-2;k)=q^{2k}+1.$$

This conjecture concerns the extremal size of constant-dimension codes with parameters $(2k,2k-2;k)$ and is motivated by the known bound in the case $A_2(8,6;4)$. The source gives only tiny numerical evidence and presents the claim as a conjecture, so its general validity remains open.