Alon–Wormald–Zhou conjecture on packing two Kneser graphs

Let $m>w\ge 1$. The Kneser graph $\operatorname{KG}(m,w)$ has as vertices the $w$-element subsets of an $m$-element set, with two vertices adjacent when the corresponding subsets are disjoint. Two copies of this graph are said to pack into a graph on the same vertex set when their edge sets can be realized as edge-disjoint copies. Writing $N=\binom{m}{w}$, Alon–Wormald–Zhou's packing conjecture. If two copies of $\operatorname{KG}(m,w)$ can be packed into the complete graph on $N$ vertices, then

$$w=\Omega(m).$$

In the cited work, the weaker bound $w=\Omega(m/(\log m)^2)$ was proved under the same assumption, so the conjectured linear bound remains open.