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Abstract

We present a phenomenological study of rotating, charged black holes in Einstein gravity coupled to a traceless (conformal) matter sector formed by ModMax nonlinear electrodynamics and a Kalb-Ramond two-form that spontaneously breaks local Lorentz symmetry. Starting from a family of obtained static, Schwarzschild-like solutions with a traceless Kalb-Ramond sector, we construct the stationary, axisymmetric counterpart via the Newman-Janis algorithm. The resulting Newman-Kerr-like metric depends on four intrinsic parameters: the electric charge $Q$, the ModMax nonlinearity $\gamma$, the Lorentz-violation amplitude $\ell$ and the spin $a$. We analyze horizon structure and separatrices in parameter space, derive the null geodesic equations and obtain the photon capture boundary that defines the black hole shadow. Using ray-tracing, we compute shadow silhouettes and a suite of shadow observables (areal radius, characteristic radius $R_s$, distortion $\delta$, oblateness $D$) and show how $\gamma$ and $\ell$ produce qualitatively distinct effects: $\gamma$ acts as a screening factor for the electromagnetic imprint, while $\ell$ introduces angular-dependent metric rescalings that deform shadow shape beyond simple size rescaling. We confront model predictions with EHT angular-radius measurements for M87$^*$ and Sgr A$^*$ and derive conservative bounds on the combinations of $(Q,\gamma,\ell,a)$. Our results identify an effective charge combination $Q_{\rm eff}\simeq e^{-\gamma}Q^{2}/(1-\ell)^{2}$ and demonstrate that modest $Q_{\rm eff}$ remains compatible with current EHT images while large $Q_{\rm eff}$ is progressively disfavored.