The join zero conjecture for Speyer's polynomial

For integers $r,s\geq1$, let $K_{2+r}\vee\overline{K}_{s+r}$ be the join of the complete graph $K_{2+r}$ and the edgeless graph on $s+r$ vertices. A polynomial has a zero of order $s$ at a point if that point is a root of multiplicity at least $s$.

Join zero conjecture. The polynomial $g_{K_{2+r}\vee\overline{K}_{s+r}}(t)$ has a zero of order $s$ at

$$-1-\frac1r.$$

This predicts systematic roots for Speyer polynomials of a broad family of joins. The paper presents it as an unproved formula supported by examples.