Wolf-Rayet stars (WRs) represent one of the final evolutionary stages of massive stars and are thought to be the immediate progenitors of stellar-mass black holes. Their multiplicity characteristics form an important anchor point in single and binary population models for predicting gravitational-wave progenitors. Recent spectroscopic campaigns have suggested incompatible multiplicity fractions and period distributions for N- and C-rich Galactic WRs (WNs and WCs) at both short and long orbital periods, in contradiction with evolutionary model predictions. In this work, we employed long-baseline infrared interferometry to investigate the multiplicity of WRs at long periods and explored the nature of their companions. We present a magnitude-limited ($K<9$; $V<14$) survey of 39 Galactic WRs, including 11 WN, 15 WC, and 13 H-rich WN (WNh) stars. We used the $K$-band instrument GRAVITY at the Very Large Telescope Interferometer (VLTI) in Chile. The sensitivity of GRAVITY at spatial scales of sim 1--200 milliarcseconds and flux contrast of $1<!PCT!>$ allowed an exploration of periods in the range $10^ $ d and companions down to sim 5$\,M_ We carried out a companion search for all our targets, with the aim of either finding wide companions or calculating detection limits. We also explored the rich GRAVITY dataset beyond a multiplicity search to look for other interesting properties of the WR sample. We detected wide companions with VLTI/GRAVITY for only four stars in our sample: WR 48, WR 89, WR 93, and WR 115. Combining our results with spectroscopic studies, we arrived at observed multiplicity fractions of $f^ WN obs WC obs and WNh obs The multiplicity fractions and period distributions of WNs and WCs are consistent in our sample. For single WRs, we placed upper limits on the mass of potential companions down to sim 5$\,M_ for WNs and WCs, and sim 7$\,M_ for WNh stars. In addition, we also found other features in the GRAVITY dataset, such as (i) a diffuse extended component contributing significantly to the $K$-band flux in over half the WR sample; (ii) five known spectroscopic binaries resolved in differential phase data, which constitutes an alternative detection method for close binaries; and (iii) spatially resolved winds in four stars: WR 16, WR 31a, WR 78, and WR 110. Our survey reveals a lack of intermediate- (a few hundred days) and long- (a few years to decades) period WR systems. The 200d peak in the period distributions of WR+OB and BH+OB binaries predicted by Case B mass-transfer binary evolution models is not seen in our data. The rich companionship of their O-type progenitors in this separation range suggests that the WR progenitor stars expand and interact with their companions, most likely through unstable mass transfer, resulting in either a short-period system or a merger.
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