The complete graph recurrence conjecture for Speyer's polynomial

Let $K_n$ be the complete graph on $n$ vertices, let $g_{K_n}(t)$ denote its Speyer polynomial, and let $g'_{K_{n-1}}(t)$ denote the derivative of $g_{K_{n-1}}(t)$ with respect to $t$. The complete graph recurrence conjecture. For every integer $n\geq 5$,

$$g_{K_n}(t)=(n-2+t(n-3))g_{K_{n-1}}(t)+t(1+t)g'_{K_{n-1}}(t)+(1+t)(t-n+3)g_{K_{n-2}}(t).$$

The recurrence was verified computationally for all $n\leq 40$ in the paper, but no proof is supplied, so it remains open.