Polynomiality conjecture for Lecture Hall polynomials

Let $\ell_i$ denote the $i$-th Lecture Hall polynomial, constructed in the paper as a Laurent polynomial depending on $y_1,\dots,y_i$. Polynomiality conjecture for Lecture Hall polynomials. Each Lecture Hall polynomial $\ell_i$ is a polynomial. The conjecture is supported by computational evidence and is needed for the proposed algebra $A_n=\mathbb{Q}[\ell_1,\dots,\ell_n]$ to be a subalgebra of an ordinary polynomial ring rather than merely a Laurent polynomial ring.