Immersion number conjecture for generalized Mycielski graphs of complete graphs

For integers $m\geq 3$, let $K_{m+1}$ be the complete graph on $m+1$ vertices, let $\mu_m(K_{m+1})$ be its $m$-Mycielskian, and let $\operatorname{im}(G)$ denote the largest $t$ such that $G$ has a $K_t$-immersion. Immersion number conjecture.

$$\operatorname{im}\bigl(\mu_m(K_{m+1})\bigr)=2m+1.$$

The conjecture arises from examples showing unusually large immersion numbers for generalized Mycielski graphs, including $\operatorname{im}(\mu_3(K_4))=7$ and $\operatorname{im}(\mu_4(K_5))=9$. The source does not provide a resolution.