Thomas–Yong–Mihalcea Grothendieck positivity inequality

Let $\mathcal P$ denote the set of partitions, let $\widetilde{G}_{\lambda}$ be the modified stable Grothendieck symmetric function, and let $\leqslant_{\operatorname{g}}$ denote Grothendieck positivity. Thomas–Yong–Mihalcea's Grothendieck positivity conjecture. For all partitions $\lambda,\mu\in\mathcal P$, one has

$$\widetilde{G}_{\lambda}\\,\widetilde{G}_{\mu}\leqslant_{\operatorname{g}}\widetilde{G}_{\lambda\vee\mu}\\,\widetilde{G}_{\lambda\wedge\mu}.$$

The corresponding Schur-positivity inequality is proved in the paper, whereas the stronger Grothendieck-positivity assertion remains open.