  Radio emission from pulsars can be used to map out their distances through
dispersion measure (DM), which quantifies the amount of radio pulse dispersion.
However, this method relies on accurately modelling the free electron density
in the line of sight. Here, we present a detailed study of the multiwavelength
emission from PSR J1720$-$0534, a black widow compact binary millisecond pulsar
discovered in 2021, which the latest electron density model of the Galaxy (Yao
et al. 2017) places at only 191 pc. We obtained and analysed deep
multiwavelength observations in the $\gamma$-ray (Fermi-Large Area Telescope,
2008-2022), optical (Las Cumbres Observatory, 2.7 h), near-infrared (Nordic
Optical Telescope, 3.5 h), and X-ray (Swift-X-Ray Telescope, 10 ks) bands. We
found no significant detection of $\gamma$-ray, optical, near-infrared, or
X-ray counterparts around the radio-timing position of PSR J1720$-$0534, which
we thus nickname 'the invisible black widow'. Employing the most constraining
near-infrared limit ($J>23.4$ mag), we established a lower limit on the source
distance, $d>1.1$ kpc, assuming conservative properties for the black widow
companion star. This distance lower limit differs drastically (by a factor of
more than 5) from the Yao et al. DM distance estimate. We attribute this
difference to the inclusion in the Yao et al. model of a large and dense
component towards the North Polar Spur. Considering our results and recent
parallax distances to other pulsars in this direction, we argue that such a
local and large component in the electron density model of the Galaxy is
unnecessary.
