The unlabelled weighted graph conjecture for horizontal-strip LLT polynomials

Let $\bm\lambda$ and $\bm\mu$ be horizontal strips, and let $\Pi(\bm\lambda)$ and $\Pi(\bm\mu)$ be their weighted graphs. If

$$\Pi(\bm\lambda)\cong\Pi(\bm\mu)$$

as weighted graphs, then the associated LLT polynomials satisfy

$$G_{\bm\lambda}(\bm x;q)=G_{\bm\mu}(\bm x;q).$$

This conjecture extends the triangle-free result to arbitrary weighted graphs arising from horizontal strips: the LLT polynomial should be determined by the unlabelled weighted graph. No resolution is supplied in the source.