Circular LLT chromatic Schur-positivity difference conjecture

Let $\mathbf{a}$ be a circular area sequence, and let $\mathrm{G}_{\mathbf{a}}(\mathbf{x};q+1)$ and $\mathrm{X}_{\mathbf{a}}(\mathbf{x};q)$ denote the corresponding LLT polynomial and chromatic symmetric function. Circular LLT–chromatic difference conjecture. The difference

$$\mathrm{G}_{\mathbf{a}}(\mathbf{x};q+1)-\mathrm{X}_{\mathbf{a}}(\mathbf{x};q)$$

is Schur-positive. The claim is motivated by computer experiments and remains open.