Hausman test VS Mundlak model; choosing between fixed-effects and random-effects

I am using panel data and I have to choose between fixed-effect and random-effect models.

I run the Hausman test, the H0 (i.e., the difference in the coefficients from the two models is not systematic) is rejected. Thus, I should use the fixed-effect.

I also run a second test, which is based on the Mundlak model.
So, I run the Mundlak regression (i.e., a random-effect model where I also add the individual means of the time-varying characteristics) and I run an F-test on the extra coefficients (the coefficients of the individual means of the time-varying characteristics).
I cannot reject the null hypothesis. Thus, I should use the random-effect model (although also the fixed-effect is unbiased, the random effect is more efficient).

I am puzzled. I test the same hypothesis in two different ways, obtaining two opposite results.
I am not sure what I should do...
Did I misinterpreted something?
Do you have a possible explanation for the two tests giving different results?