Havet–Horsch–Rambaud lexicographic-product conjecture for inversion diameter

Let $G$ be a graph and let $t$ be a positive integer. Denote by $G[\overline{K_t}]$ the lexicographic product obtained by replacing each vertex of $G$ with an independent set of size $t$, and write $I(G)$ for the inversion graph and $\operatorname{diam}(I(G))$ for its diameter.

Havet–Horsch–Rambaud's lexicographic-product conjecture. For every graph $G$ and every positive integer $t$,

$$\operatorname{diam}(I(G[\overline{K_t}]))\le t\cdot\operatorname{diam}(I(G)).$$

The conjecture predicts that lexicographic blow-ups by independent sets increase inversion diameter by at most the blow-up factor. It is presented as a proposed general bound, and no resolution is supplied in the source.