Generalized $b$-conjecture for non-oriented constellations

Let $k\geq 1$ and let $\lambda,\mu^0,\ldots,\mu^k$ be partitions of size $n\geq 1$. A connected rooted $k$-constellation with profile $(\lambda,\mu^0,\ldots,\mu^k)$ is denoted by $(\mathbf{M},c)$, where $c$ is the root. Generalized $b$-conjecture. There exists a function $\nu$ on the connected rooted constellations with profile $(\lambda,\mu^0,\ldots,\mu^k)$, taking non-negative integer values, such that $\nu(\mathbf{M},c)=0$ if and only if $(\mathbf{M},c)$ is orientable, and

$$h^\lambda_{\mu_0,\ldots,\mu_k}(b)=\sum_{(\mathbf{M},c)}b^{\nu(\mathbf{M},c)},$$

where the sum runs over rooted connected $k$-constellations with profile $(\lambda,\mu^0,\ldots,\mu^k)$. This is the constellation counterpart of the generalized Matching-Jack conjecture and is equivalent to the generalized positivity conjecture in the setting of the paper. The case $k=1$ remains open, although several special cases and partial results are known.