Linear odd-colouring bound for complete-graph immersions

Let $G$ be a graph and let $G\xrightarrow{\operatorname{odd}} K_{ct}$ mean that $G$ admits a $t'$-oddomorphism to $K_{ct}$, as defined in the source. Linear immersion conjecture. There exists an absolute constant $c$ such that if

$$G\xrightarrow{\operatorname{odd}} K_{ct},$$

then $G$ contains a $K_t$-immersion.

The conjecture would improve the established bound $\binom{t}{2}(7t+7)$ to a bound linear in $t$. The source does not state whether it has been resolved.