Pak's conjecture on the size-Ramsey number of long subdivisions

Given a graph $H$ and a function $c3:E(H)\to\mathbb{N}$, the subdivision $H^{\sigma}$ is obtained by replacing each edge $e\in E(H)$ with a path of length $\sigma(e)$. For a graph $G$ and positive integer $k$, let $\widehat{R}(G,k)$ denote its multicolor size-Ramsey number, and let $\Delta(H)$ denote the maximum degree of $H$. Pak's conjecture. For every $k,D\in\mathbb{N}$ there exist $C,L>0$ such that if $H$ is a graph with $\Delta(H)\leq D$ and

$$\sigma(e)\geq L\log(|V(H^{\sigma})|)$$

for all $e\in E(H)$, then

$$\widehat{R}(H^{\sigma},k)\leq C|V(H^{\sigma})|.$$

Pak posed this conjecture for long subdivisions of bounded-degree graphs, where every subdividing path has logarithmic length in the number of vertices. It predicts a linear multicolor size-Ramsey bound; the supplied text gives no information about whether the conjecture has been resolved.