Column-sum conjecture for inverse double-dimer matrices

Let $M^{-1}$ be the inverse matrix indexed by paths, and let $\mu$ be a column index. For each chord $c$ under $\mu$, let its height be the number specified by the diagram, and let $f_{\lambda/\mu}(q)$ denote the associated generating polynomial. Column-sum conjecture. The absolute values of the entries in any column $\mu$ of $M^{-1}$ sum to the product of the chord heights under $\mu$, with $q$-analogue

$$\sum_\lambda f_{\lambda/\mu}(q)=\prod_{\text{chord }c\text{ of }\lambda}(\text{height of }c)_q.$$

The source presents this alongside the row-sum formula and does not give a proof; the notation and product indexing should be checked against the accompanying figure.