A conjectural formula for the first coefficient of the maximum likelihood degree polynomial

Let $m\geq 2$, and let $\Lambda_m$ be the object whose coefficients are $\lambda_1,\dots,\lambda_{m-1}$. Conjectural formula. For $m\geq 2$, the first coefficient satisfies

$$\lambda_1=(-1)^m(3^m-2^{m+1}+1).$$

This is one of the closed-form formulas proposed for the maximum likelihood degrees of mixtures of independence models; its status is not resolved in the supplied text.