On the image of higher signature maps

Given a smooth variety $X$ over the field $\mathbb{R}$ of real numbers and a line bundle $\mathcal{L}$ on $X$ with associated topological line bundle $L=\mathcal{L}(\mathbb{R})$, we study the quadratic real cycle class map ${\tilde{\gamma }}_{\mathbb{R}}^c:{\widetilde{\mathrm{C}\mathrm{H}}}^c(X,\mathcal{L})\to {\mathrm{H}}^c(X(\mathbb{R}),\mathbb{Z}(L))$ from the $c$-th Chow-Witt group of $X$ to the $c$-th cohomology group of its real locus $X(\mathbb{R})$ with coefficients in the local system $\mathbb{Z}(L)$ associated with $L$. We focus on the cases $c\in \{0,d-2,d-1,d\}$ where $d$ is the dimension of $X$ and we formulate a precise conjecture on the image of ${\tilde{\gamma }}_{\mathbb{R}}$ in terms of the exponents of its cokernel that is corroborated by the results obtained in those codimensions.