Marginal $b$-conjecture for map enumeration

For all $n,f\geq 1$ and partitions $\nu\vdash 2n$, let $c_{\mu,\nu,n}(b)$ denote the coefficient associated with maps having $n$ edges and vertex and face profiles $\mu$ and $\nu$. Let $\vartheta(\mathfrak{m})$ measure the non-orientability of a possibly non-orientable map $\mathfrak{m}$, with $\vartheta(\mathfrak{m})=0$ if and only if $\mathfrak{m}$ is orientable. Marginal $b$-conjecture. For all $n,f\geq 1$ and $\nu\vdash 2n$,

$$\sum_{\substack{\mu\vdash 2n\l(\mu)=f}}c_{\mu,\nu,n}(b)=\sum_{\mathfrak{m}}b^{\vartheta(\mathfrak{m})},$$

where the sum on the right is over maps with $n$ edges and vertex and face profiles prescribed by $\mu$ and $\nu$. This marginal identity refines the enumeration by a statistic of non-orientability; the supplied text gives no resolution status.