The third match says that "Fixed effects extensions to hurdle, finite mixture, zero-inflated models are currently not available".
An alternative to "classical" FE-regression would be a mixed model. Unlike often stated, mixed models can indeed deal with heterogeneity bias and even incorporate between-effects and time-constant predictors, and even account for variation of level-1-predictors between groups (i.e. random slopes). These kind of models are also called "complex random effects within-between models" .
You can then use the glmmTMB package, which lets you fit negative binomial mixed models with zero-inflation.
There is an example how to do this in lme4, but you can easily extend this to glmmTMB: https://easystats.github.io/parameters/articles/demean.html
The above linked example also includes visual examples and some references for further reading. Imho, mixed models clearly outperform classical FE-models, so I would go this route. And glmmTMB is a great package for mixed models with many different error term distributions / families.
 Bell, Andrew, Malcolm Fairbrother, and Kelvyn Jones. 2019. “Fixed and Random Effects Models: Making an Informed Choice.” Quality & Quantity 53: 1051–74. https://doi.org/10.1007/s11135-018-0802-x.