  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.
