Stembridge's conjecture on total nonnegativity of monomial trace immanants

Let $n$ be a positive integer, let $\lambda\vdash n$ be a partition, and let $\phi^\lambda$ denote the corresponding monomial trace. For an $n\times n$ matrix $x$, write $\operatorname{Imm}_{\phi^\lambda}(x)$ for its monomial trace immanant. Stembridge's conjecture. For every partition $\lambda\vdash n$, the polynomial

$$\operatorname{Imm}_{\phi^\lambda}(x)$$

is totally nonnegative. The preceding corollary does not apply because $\phi^\lambda(\operatorname{inc}(P))<0$ for some partitions $\lambda$ and posets $P$; the conjecture asserts total nonnegativity nevertheless and is attributed in the source to Stembridge.