Nonnegative-coefficient property of the parabolic F-triangle

Let $n>0$ and let $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_r)$ be a composition of $n$ into $r$ parts. Let $F_{\alpha}(x,y)$ be the rational function defined from the H-triangle by

$$F_{\alpha}(x,y)=x^rH_{\alpha}\left(\frac{x+1}{x},\frac{y+1}{x+1}\right).$$

F-triangle conjecture. The rational function $F_{\alpha}(x,y)$ is a polynomial with nonnegative integer coefficients if and only if $\alpha$ has at most one part exceeding $1$.

The source presents this as a computational pattern and relates it to the preceding conjecture characterizing when the corresponding H-triangle identity holds. No proof or resolution is supplied.