  Over the last few years, low- and high-redshift observations set off a
tension in the measurement of the present-day expansion rate, $H_0$. Adding to
the riddle, observational data from the Planck mission point to a $3.4\sigma$
evidence for a closed universe, further challenging the $\Lambda$CDM
concordance model of cosmology. Recently, a direct-observational test has been
proposed to discriminate effects of the spatial curvature in the cosmological
model. The test is based on the fundamental distance--flux--redshift relation
of the luminosity distance modulus, $\Delta \mu$. We reexamine the outcomes of
this test and show that achieving the required $\Delta \mu$ sensitivity to
discriminate among cosmological models is materially far more challenging than
previously thought. Armed with supernova type Ia (SN Ia) data, calibrated using
Cepheid measured distances, we apply the test to archetypal spatially non-flat
models that ameliorate the $H_0$ tension and show that the $3\sigma$ contour of
$\Delta \mu$ predicted by these models overlaps the 68\% CL SN Ia residuals
with respect to $\Lambda$CDM. This implies that the spatial curvature remains
insensitive to local $H_0$ measurements from the Cepheid distance ladder.
