Row-sum conjecture for inverse double-dimer matrices

Let $M^{-1}$ be the inverse matrix indexed by paths, and let $\lambda$ be a row of order $n$. For a chord $c$ of $\lambda$, write $|c|$ for its length, and let $f_{\lambda/\mu}(q)$ denote the associated generating polynomial; write $n!_q$ and $|c|_q$ for the corresponding $q$-analogues. Row-sum conjecture. For any row $\lambda$ of $M^{-1}$, the absolute values of its entries sum to a divisor of $n!$, more precisely

$$\sum_\mu q^{|\lambda/\mu|/2}f_{\lambda/\mu}(q^{1/2})=\frac{n!_q}{\prod_{\text{chord }c\text{ of }\lambda}|c|_q}.$$

At $q=1$, this gives the stated row sum $n!/\prod_c|c|$. The conjecture is proved for the first row, where the row sum is $n$, while the general formula remains unproved in the source.