The refined coefficient conjecture for multivariate diagonal harmonics

Fix a partition $\lambda\vdash n$, and let $c_\lambda(q,r)$ be the coefficient of $e_\lambda$ in the Frobenius characteristic $\Phi_{n,r}(q)$. Write $\lambda_i=0$ for $i$ larger than the length of $\lambda$, and set

$$m=\sum_{i=1}^n(n-i+1)(\lambda_i-1).$$

Here $m$ is the maximal area of a Dyck path with connected north steps of type $\lambda$. Refined coefficient conjecture. The polynomial $c_\lambda(q,r)$ is a polynomial in $r$ with a $q$-positive expansion in the binomials $\binom{r-2}{*}$. Moreover,

$$c_\lambda(q,r)=\binom{r-2}{m}+(q+b)\binom{r-2}{m-1}+\text{lower terms},$$

where $b$ is a positive integer. The conjecture was verified by computer up to $n=6$ and refines the proposed $e$-positivity phenomenon for multivariate diagonal harmonics.