We present a detailed analysis for a subset of the high resolution (~35 mas, or 5 au) ALMA observations from the Disk Substructures at High Angular Resolution Project (DSHARP) to search for faint 1.3 mm continuum emission associated with dusty circumplanetary material located within the narrow annuli of depleted emission (gaps) in circumstellar disks. This search used the Jennings et al. (2020) $tt{frank}$ modeling methodology to mitigate contamination from the local disk emission, and then deployed a suite of injection-recovery experiments to statistically characterize point-like circumplanetary disks in residual images. While there are a few putative candidates in this sample, they have only marginal local signal-to-noise ratios and would require deeper measurements to confirm. Associating a 50% recovery fraction with an upper limit, we find these data are sensitive to circumplanetary disks with flux densities $gtrsim 50-70$ $mu$Jy in most cases. There are a few examples where those limits are inflated ($gtrsim 110$ $mu$Jy) due to lingering non-axisymmetric structures in their host circumstellar disks, most notably for a newly identified faint spiral in the HD 143006 disk. For standard assumptions, this analysis suggests that these data should be sensitive to circumplanetary disks with dust masses $gtrsim 0.001-0.2$ M$_oplus$. While those bounds are comparable to some theoretical expectations for young giant planets, we discuss how plausible system properties (e.g., relatively low host planet masses or the efficient radial drift of solids) could require much deeper observations to achieve robust detections.
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