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The connection between the escape fraction of ionizing photons ($f_{\rmesc}$) and star-formation rate surface density ($\Sigma_{\rm SFR}$) is a keyinput for reionization models, but remains untested at high redshift. Weanalyse 35 z~3 galaxies from the Keck Lyman Continuum Survey (KLCS) covered bydeep, rest far-UV spectra of the Lyman continuum (LyC) and high-resolution HSTV$_{606}$ imaging, enabling estimates of both $f_{\rm esc}$ and rest-UV sizes.Using S\'ersic profile fits to HST images and spectral-energy distribution fitsto multi-band photometry, we measure effective sizes and star-formation ratesfor the galaxies in our sample, and separate the sample into two bins of$\Sigma_{\rm SFR}$. Based on composite spectra, we estimate <$f_{\rm esc}$> forboth $\Sigma_{\rm SFR}$ subsamples, finding no significant difference in<$f_{\rm esc}$> between the two. To test the representativeness of the KLCS HSTsample and the robustness of this result, we attempt to recover thewell-established correlation between $f_{\rm esc}$ and Ly$\alpha$ equivalentwidth. This correlation is not significant within the KLCS HST sample,indicating that the sample is not sufficient for correlating $f_{\rm esc}$ andgalaxy properties such as $\Sigma_{\rm SFR}$. We perform stacking simulationsusing the KLCS parent sample to determine the optimal sample size for robustprobes of the $f_{\rm esc}$-$\Sigma_{\rm SFR}$ connection to inform future HSTLyC observing programs. For a program with a selection independent of ionizingproperties, >= 90 objects are required; for one preferentially observingstrongly-leaking LyC sources, >= 58 objects are required. More generally,measuring the connection between $f_{\rm esc}$ and $\Sigma_{\rm SFR}$ requiresa larger, representative sample spanning a wide dynamic range in galaxiesproperties such as $\Sigma_{\rm SFR}$.

Searching for the connection between ionizing-photon escape and the surface density of star formation at z~3