Positivity conjecture for the Canham-model recurrence sequence

Let $(d_n)$ be the unique sequence of rational numbers with initial terms

$$(d_0,\ldots,d_6)=\left(72,1932,31248,\frac{790101}{2},\frac{17208645}{4},\frac{338898609}{8},\frac{1551478257}{4}\right)$$

satisfying the explicit order-seven linear recurrence

$$\sum_{i=0}^7 r_i(n)d_{n+i}=0,$$

where each $r_i(n)\in\mathbb{Z}[n]$ is defined by the recurrence in the appendix. Positivity conjecture. Every term of the sequence $(d_n)$ is positive. This positivity is the sequence-theoretic statement to which Yu and Chen reduce the genus-one part of their Canham-model uniqueness conjecture; the source does not provide a resolution.