Flagged Grothendieck characterization conjecture for Lascoux polynomials

Let $\lambda$ be any partition and let $w\in S_n^{\lambda}$. A flagged Grothendieck polynomial is understood in the sense used in the paper, and $w$ is 312-avoiding when it has no subsequence with relative order $312$. Flagged Grothendieck characterization conjecture. The Lascoux polynomial $L_{w\lambda}(\mathbf{x};\beta)$ is a flagged Grothendieck polynomial if and only if $w$ is 312-avoiding. This is the K-theoretic analogue of the corresponding characterization for Demazure characters at $\beta=0$; the conjecture seeks to identify exactly which Lascoux polynomials admit flagged Grothendieck descriptions.