Deep VLBI constraints on compact radio cores in four ultraluminous X-ray sources

We present high-sensitivity Very Long Baseline Interferometry (VLBI) observations of four ultraluminous X-ray sources (ULXs): Holmberg II X-1, IC 342 X-1, NGC 6946 X-1, and NGC 925 X-1. No compact emission was detected on milliarcsecond scales, with rms noise levels reaching approximately 5--20 $\mu$Jy. The corresponding $5\sigma$ flux density upper limits reach $\sim 26\,\mu\mathrm{Jy}$, implying radio luminosity limits $L_{\rm R} \lesssim 2 \times 10^{33}\,\mathrm{erg\,s^{-1}}$. This disfavors any persistently bright hard-state-like compact core at our sensitivity level. The previously reported VLBI core in Holmberg II X-1 exhibits significant long-term variability, broadly consistent with an overall decline over the past decades. This behavior is consistent with emission from optically-thin ejecta undergoing adiabatic expansion. The VLBI non-detections may reflect intrinsically weak/intermittent compact emission, and/or low--surface--brightness structure that is resolved out by VLBI, and/or absorption/propagation effects such as free--free absorption in dense, ionized winds.