Bergeron's e-positivity conjecture for multivariate diagonal harmonics

Let $\mathrm{DH}_{n,r}$ denote the multivariate diagonal harmonics space, and let $\Phi_{n,r}(q)$ be the Frobenius characteristic of its character, with parameter $q_r$. A symmetric function is $e$-positive if its expansion in the elementary symmetric-function basis has nonnegative coefficients. Bergeron's e-positivity conjecture. For every fixed $n$ and $r$, the Frobenius characteristic $\Phi_{n,r}(q)$ is $e$-positive when $q_r=1$. The conjecture is presented as evidence for a Hopf-chain description of multivariate diagonal harmonics and is open in the source.