Gauge homotopy equivalence of the homotopy inverse composite

Let $h_=h\Psi$ be the element defined in the last remark before the appendix of Dotsenko, Shadrin, and Vallette (2016). Let i_ and p_ be the $_\alpha$-morphisms whose composite is i_\circledcirc p_, and let $\mathrm{id}$ denote the identity $_\alpha$-morphism. Gauge homotopy equivalence conjecture. The element $h_=h\Psi$ is a gauge homotopy equivalence of $_\alpha$-morphisms between i_\circledcirc p_ and $\mathrm{id}$. This conjecture concerns the economical gauge-homotopy notion for infinity-morphisms introduced in the paper; its validity is not resolved in the supplied text.