Classical Cepheids are radially pulsating stars that enable important tests of stellar evolution and play a crucial role in the calibration of the local Hubble constant. ℓ Carinae is a particularly well-known distance calibrator, being the closest long-period (P ~ 35.5 d) Cepheid and subtending the largest angular diameter. We have carried out an unprecedented observing program to investigate whether recently discovered cycle-to-cycle changes (modulations) of ℓ Carinae's radial velocity (RV) variability are mirrored by its variability in angular size. To this end, we have secured a fully contemporaneous dataset of high-precision RVs and high-precision angular diameters. Here we provide a concise summary of our project and report preliminary results. We confirm the modulated nature of the RV variability and find tentative evidence of cycle-to-cycle differences in ℓ Car's maximal angular diameter. Our analysis is exploring the limits of state-of-the-art instrumentation and reveals additional complexity in the pulsations of Cepheids. If confirmed, our result suggests a previously unknown pulsation cycle dependence of projection factors required for determining Cepheid distances via the Baade-Wesselink technique.