This paper investigates state estimation for continuum robotic fingers in feature-sparse and dynamic in-hand manipulation environments. Continuum fingers, inspired by continuum robots, offer enhanced flexibility and wider reachable workspace compared with conventional rigid-link fingers to enable grasping and manipulation tasks. However, they lack encoder-based joint angle measurements, making it difficult to determine fingertip positions, particularly under external forces during contact. This limitation hinders precision grasping and prevents the full exploitation of their high dexterity. To address this challenge, we developed a simultaneous localization and mapping framework for continuum fingers using proximity sensors. Unlike conventional simultaneous localization and mapping that assumes feature-rich environments, grasping scenarios present feature-sparse conditions with limited environmental information. We propose an estimator that fuses proximity sensing with a constant-curvature kinematic prior by replacing encoder angles with virtual joint angles. The key idea is to leverage the designed in-hand elements, namely opposing fingers and the palm, as stable reference geometry. Simulations demonstrate that the proposed estimator outperforms a kinematics-only baseline by suppressing bias and reducing position error. Three-dimensional contoured palms enhance observability, with a composite wavy palm yielding the smallest errors without temporal drift. These findings indicate that the designed in-hand geometry combined with temporal map management enables effective state estimation for continuum fingers in feature-sparse and dynamic grasping scenarios.

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