The polynomial formula conjecture for six-lozenge tilings

Let $L_{n,l}$ denote the number of lozenge tilings of the equilateral triangle of side length $n$ containing $l$ lozenges, and let $M_{n-1}$ be the polynomial appearing in the upper bound for $L_{n,l}$. For $n\ge 4$, the six-lozenge polynomial conjecture.

$$L_{n,6}=\binom{M_{n-1}}{6}-\bigg[3(n-1)^2\binom{M_{n-1}-2}{4}-\frac{1}{16}(-131088+61472n-41206n^3+69420n^2-90n^6-918n^7+153n^8-10851n^4+9828n^5)\bigg]$$ $$=\frac{1}{5120}(81n^{12}-486n^{11}-2835n^{10}+21870n^9+26775n^8-384786n^7+131751n^6+3275730n^5-3798716n^4-13254088n^3+22481984n^2+19678080n-42024960).$$

This is part of the proposed polynomial pattern for $L_{n,l}$, in which the leading upper bound has degree $2l$ and the first- and second-order corrections have degrees $2l-2$ and $2l-4$, respectively. The formula is presented as a conjectural consequence of the observed correction pattern.